LECTURES ON RANDOM MEASURES
by
Olav Kallenberg
Univepsity of
~tebopg,
SWeden
and
Univepsity of Nopth Capolina at Chapel Hill
Department of Statistias
Institute of Statistics Mimeo Series No. 963
November, 1974
LECTURES ON RANDOM MEASURES
by
Olav Kallenberg
PREFACE
Random measure theory is a new and rapidly growing branch of
probabi li ty of increasing interest both in theory and applications.
Loosely speaking, it is concerned with random quantities which can
only take non-negative values, such as e.g. the number of random variables
in a given sequence possessing a certain property, the time spent by a
random process in a certain region etc.
In the special case when these
quantities are integer valued, our random measures reduce to point processes,
which constitute an important subclass of random measures.
However, I am
convinced that general random measures are potentially just as important
from the point of view of applications.
(See e.g. Kallenberg (1973c)
for some results supporting this opinion.)
The justification of random measure theory as a separate topic lies
in the overwhelming amount of important results which are specific to
random measures in the sense that they do not carryover to general random
processes.
As a contrast, comparatively few results are specific in this
sense to the subclass of point processes.
Indeed, it will he clear from
the development on the subsequent pages that a substantial part of point
process theory, as presented e.g. in the monograph of Kerstan, Matthes
..
and Mecke (1974), carries over to the class of general random measures.
The latter may therefore be considered as a natural scope of a theory.
These lectures contain research supported in part hy the Office of Naval
Research under Contract NOOO1tl-67-I\-O~21-0002.
e
-2-
The origin of these notes is a series of lectures delivered at the
Statistics Department at Chapel Hill during the spring semester of 1974,
where I intended to collect the (in my opinion) central facts about random
measures from the scattered literature in the field.
During the course of
my lectures, however, I often elaborated new ideas and approaches, and as
a result I found many improvements of known theorems and proofs, and also
a large number of new results.
The present notes should therefore be
regarded both as a survey of known results and as an original contribution
to the research in the area.
Here follows
a summary of the contents and an indication about
the nature of the most important novelties.
Section 1 is introductory.
After having set up the general framework, we demonstrate how the basic
point processes may be defined by a simple mixing procedure.
Here and
in subsequent sections, the Laplace transform plays the role of a powerful
general tool.
In Section 2, the main purpose is to extend the classical
simplicity criteria for point processes into a general theory of atomic
properties for arbitrary random measures.
Section 3 is devoted to the
question of uniqueness of random measure distributions.
In particular,
it is shown that the distribution of a simple point process or diffuse
~
random measure
=
Ee-t~B
is uniquely determined by the set function
for bounded Borel sets
Note that, as
t
+ 00,
~t(B) +
B, where
P{~B =
O}
into a well-known one for point processes.
t
>
0
~t(B) =
is arbitrary but fixed.
and our result turns formally
In Section 4, we derive the
-3-
corresponding convergence criteria.
A novelty here is a relationship
between the notions of convergence based on the weak and vague topologies
respectively.
The idea of tightness leads in Section 5 to a simple
approach to the existence theorems of Harris and others.
Sections 6 and 7 are devoted to infinite dlivisibility and the limit
theorems for null-arrays.
divisible random measure
From a simple representation of an infinitely
l:;,
we are able in Section 6 to draw interesting
conclusions about the relationship between the properties of
l:;,
itself
and its canonical spectral measure; in Section 7 we show among other things
how the above-mentioned uniqueness theorem in Section 3 leads in particular
cases to improved convergence criteria for null-arrays.
In Section 8 on
compounding and thinning of point processes, the known results for the
case when the compounding variables
f3
n
tend to zero in distribution
are accompanied by an analogeous theory for the complementary
case when
no subsequence tends to zero.
Symmetrically distributed random measures are examined in Section
9, and in particular we dwell on the case when the random measures are
at the same time infinitely divisible.
The peak of our treatment is
the characterization of the class of canonical measures which can arise
in the corresponding spectral representations.
After a general discussion
of Palm distributions in Section 10, containing improvements of known
uniqueness and continuity results, we finally enter in Section 11 into
the fascinating but difficult subject of characterizations by factorization
and invariance properties of the Palm distributions.
As for the factorization
e
-4-
properties, we are able to give a very simple proof of the KerstanKummer-Matthes theorem, while a modification of the original proof
yields a radically strengthened version of this result.
I hope that the technicalities of the above presentat,ion, mainly
intended for specialists, will not discourage newcomers to the field.
Great efforts have been made to provide a smooth approach to the subject,
and wherever I have relied on "known" facts, detailed references are given.
Still some requisites may prove helpful, including some basic knowledge of
measures on locally compact spaces (to be found in Bauer (1972)), and of
convergence of probability measures (as presented in Billingsley (1968),
Chapter 1).
Finally, I would like to express my thanks to my audience in Chapel
Hill, and in particular to Professor M. R. Leadbetter, for their interest
in this project and for several corrections and improvements of my original
draft.
I am also grateful to the Institute of Statistics there for allowing
me to include these lecture notes in their Mimeo Series, and to Mrs. Melinda
Beck for her careful typing.
G~teborg
in August 1974
O.K.
-5-
CONTENTS
1.
Basic Concepts
6
2.
Atomic Propert ies
3.
Uniqueness ....................•..................... :;0
4.
Convergence ..........•.......................•...... :;9
5.
Existence
6.
Null-Arrays and Infinite Divisibility ..........•.... 66
7.
Further Results for Null-Arrays
87
8.
Compound and Thinned Point Processes
99
9.
Symmetrically Distrihuted Random Measures .....•.... 113
18
57
10.
Palm Distributions
1:16
11.
Characterizations Involving Palm Distributions
151
.
-6-
1.
BASIC CONCEPTS
We could confine ourselves to random measures defined on the real
R or on some Euklidean space
line
kEN
= {1,2,
... J , but it
turns out that we can choose a much more general framework without any
essential changes in notation or proofs.
abstract topological space
Hence we consider instead an
S which is assumed to be locally compact,
second countable and Hausdorff, (see e.g. Simmons (1963)).
a suitable metric
metric space.
p
,
we can convert
S
By choosing
into a separable and complete
This possibility is often useful, but it is important to
realize that our definitions or results never depend on the actual choice
of
p
•
Let
B be the Borel a-algebra in S, i.e. the a-algebra generated
by the class of open sets, and let
(By definition, a set
B be the ring of all bounded B-sets.
B is bounded whenever its closure
In general, this is stronger than metric boundedness.)
for the class of !-measurable functions
Fc
=
Fc (S)
f: S
R
+
+
=
B is compact.
Write
[0 00)
"
F
= F(S)
and let
denote its subclass of continuous functions with compact support.
To simplify statements, we introduce three types of subclasses of
By a DC-Bemiring (D
I
c
for dissecting,
B.
C for covering) we mean a semi-ring
B (see e.g. Kolmogorov and Fomin (1970) for the definition) with the
property that, given any
covering of
B
E
Band
£ >
0 , there exists some finite
B by I-sets of maximal diameter
A DC-ring is a ring with the same property.
exist and may be chosen countable.
<
£
in some fixed metric.
Note that such classes always
(We may e.g. take the ring generated
-7-
B.)
by some countable base in
On
R, typical DC-semirings and -rings
consist of finite intervals and interval unions respectively, hence our
notation.
B
E
C c B is covering if any set
Finally, we say that a class
B can be covered by finitely many sets in C .
Note that the notions of DC-semiring and --ring are purely topological
and do not depend on the choice of metric
and
are two metrizations of
pI
arbitrary.
is compact.)
I
G
G
=
E: I
in
1\
G
< EI
p
.
E'
>
0
such that
(G
B .
c
= S\G
p'
p(s,t)
Thus any covering of
(B, (;c)
-
~
E:
Band
B
(Note that any countable
and
whenever
< E
< E.
and
N
= N(S)
of
E:
G with
<E" =
(;) is included
B
M = M(S)
is compact while
of Borel measures
Z+ -valued measures
(Z +
=
G
]J
E" >
on
(S, B)
B, and further the
{O, 1, ... })
]J +]JB
of
M or
N into
R
+
measurable for each
Note that, by monotone convergence, the integral
lIere and below,
B denotes the closure of
B.
0
is open.
Let
M
N be the smallest a-algebras in M and N respectively making
the mapping
I
onto itself
It remains to verify that
which are locally finite, i.e. take finite values on
subclass
s,t
with p I -diameters
which follows easily from the facts that
We now introduce the class
G
B
p, and in particular we
denoting the complement of
and hence has p-diameters
p
be
0
>
E:
is compact, the identity mapping of
is uniformly continuous with respect to
pI (s, t)
B
B, and that a finite subcDvering exists since
Since
may choose some
In fact, suppose that
S, and let
Choose a bounded open set
B covers
base in
p.
B
E
B
-8-
is then automatically measurable for any
f
E
f
E
F. More generally, assuming
f~
F to be bounded with compact support and defining the measure
(f~)B
JBf(S)~(dS)
=
measurable.
,
B , it is seen that the mapping
E
B
E
B
B, being equal to
(As usual,
1
on
= lB~
B~
In particular, the restriction
measurable for every
the set
B
of
~ ~ f~
is
to
B is
~
by
lB denotes the indicator of
B and
0
elsewhere.)
LEMMA 1.1. The a-aZgebras M in M and N in N are both generated
by the mappings
I
E
r
~ ~ ~f
,
f
E
reB.
for any fixed DC-semiring
~
Proof.
The same argument applies to both
consider
M
f
Let
E
By the definition of
c
~ ~ ~B
mapping
M and
M' be the a-algebra in
F
N , so let us e.g.
M generated by the mappings
M,
is M'-measurable for every
we may choose open bounded sets
Gl ,G 2 , ...
we have to prove that the
B
B.
E
such that
lemma (cf. Simmons (1963)) there exist functions
n EN, so for any
~
~B = ~lB ~ ~fn $ ~lG
n
Hence
~f
n
~f
for
+
n
n
E
~B
~ ~ ~I ,
Fc ,and aZso by the mappings
If
B is compact,
Gn + B
f l ,f , ...
2
E
By Urysohn's
Fc
EM,
= ~Gn +
, and the M'-measurability of
~B
~B
.
follows from that of
N
We next consider an arbitrary fixed compact set
v=
{B
E
B:
~ ~ ~(B
n C)
such that
C and define
is M'-measurable} .
-9-
V
Then
is clearly closed under monotone limits and proper differences
and it contains
S.
V
On the other hand,
contains the class
compact sets, which is closed under finite intersections.
monotone class theorem
generated by
M'
that
But
C.
B
for every
E
2
B.
C of
Hence by Dynkin's
(cf. Bauer (1974)), V contains the a-algebra
a(C)
Since
B , and so
=
Jl
Jl(B n C)
-+
a(C)
is M'-measurable
C was an arbitrary compact set, this proves
M
=
The assertion involving
instead of
f
contained in
I
is proved by a similar argument, but
we consider a finite covering of
n
G
Since
n
I
B by I-sets
is a semiring, we may take the
I n1 , ... ,Ink
I
.
nJ
to be
disjoint, and then
JlB
~
Jl u I nj =
j
proving that
LEMMA 1.2.
Proof.
I·J Jl 1nJ.
-)-
Ij
JlI .
nJ
~
JlG
n + JlB ,
This completes the proof.
JlB
N EM.
Let
I c B be a countable DC-semiring, and let
Jl
be a fixed
measure in the set
M = {Jl
For any fixed finite union
V
Then
2
D
ThoUI~h
{B
M: Jll
E
U
E
E
Z+ , I
I} .
E
of I-sets, we further define
B: Jl(B n U)
E
Z }
-I-
.
lS closed under monotone limits and proper differences, and it
commonly ascribed to Oynkln, this theorem is actually due to
Si erpinski, Fund. Math .
..!2
(l92R).
-10-
contains
S since
U may be written as a disjoint union of I-sets.
On the other hand, it is easily seen that
V contains
I , and
I
being
closed under finite intersections, it follows by Oykin's theorem that
B
V ~. a(I) ~
~(B n U) E Z+
B E B , we may always choose
given
~B
and we obtain
E Z+
for all
for every
U such that
B E B , proving that
~
EN.
B E B.
But for
U 0 B , so we get
Hence
MeN , and the
N=M
converse relation being obvious, we have in fact
Since clearly
M EM, this completes the proof.
Let us now consider any fixed probability space
random measure or point process
of
(n,A,p)
into
(M,M)
or
~
on
(N,N)
(n,A,p).
By a
S we mean any measurable mapping
respectively.
By Lemma 1.2, a point
process may alternatively be defined as an N-valued random measure.
any a.s. (almost surely w.r.t.
Conversely,
P) N-valued random measure coincides with a
point process except on a P-null set (i.e. on an n-set of P-measure
0) .
For these reasons, we shall make no difference in the sequel between point
processes and a.s. N-valued random measures.
random measure to take values outside
Similarly, we shall allow a
M on a P-null set.
The class of random measures is clearly closed under addition and
multiplication by non-negative random variables.
by the definition of
while
a
and
M to observe that, if
and
n are random measures
S are R+-valued random variables, the quantity
is a random variable for every
following
~
For a proof, it suffices
B E B.
a~B +
Slightly less obvious is the
SnB
-11-
L.~.B <
J J
measure iff
Proof.
on
L-J t;.J
A series
LEMMA 1.3.
00
of random measures on
outside some fixed P-null set
that
~B <
A
A
E
B E B , except on
00
~
prove that
=
~C <
Then
and since
A
C
~ E
Hence
00
L·~·
J J
~B = L.~.B
J J
is (everywhere
Next suppose
for all
C
E
C
is covering, it follows
M a.s.
is measurable, and by the definition of
from the elementary fact that
E
~
B and is therefore measure valued.
C c B is a countable base in S
that
B
BE B .
a.s. for each
First note that, by monotone convergence,
Si) a-additive on
~s itself a random
S
It remains to
M , this follows
is a random variable for every
B .
The distribution of a random measure or point process
probability measure
p~-l
on
(M,M)
(N,N)
or
N.
E denotes the "expectation" or integral w.r.t.
P.
E~
We further define the intensity
(EOB =
theorem,
E~
~
of
E(~B)
B
is the
defined by
M or
M
where
F,
E
by
E
B ,
By Fubini's
is always a measure, though it need not be locally finite.
of
We finally define the L-transform
L~(f) = Ee-F,f
Note in particular that, for any
kEN
(L
f
E
and
for Laplace) by
F .
Bl , .. , ,'\
E
B, the function
-12-
=E
LS[.r t.1 B.]
J =1 J J
eXP[-.I
t.~B.}J
J =1 J
is the elementary L-transform of the random vector
(sB , ... ,sB )
1
k
One of the most important ways to define new random measures (or
rather their distributions) is by mixing.
For a definition, let
be an arbitrary probability space and let
{sS' S
8}
E
(8,T,Q)
be a family of
random measures which mayor may not be defined on the same probability
space.
If
P{sS
is a T-measurab1e function of
E M}
we may then consider
S as a random element in
and form the mixture of
S.
(1 )
Formally,
R(M x
Note that
given
Ps- 1
S
A)
=
JAP{~S
Q , thus obtaining the
~
and the mixing
E M}
Q(dS)
ME M
A
E
T .
is then a version of the conditional distribution of
8, and that
for any measurable function
= E'E[f(~S'S)
f·. M
x
IS]
(See
8 -+ R +
Lo~ve
(1963) page 359
for a complete discussion.)
The measurability requirement above is not easy to verify.
Here
is a cqnvenient criterion:
LEMMA 1.4. Let
3
Q
R is defined by the relation
E'f(s,S)
{se '
M
M E
8 with distribution
R of the mixed random measure
joint distribution
element
with respect to
for every
8
8
Here
E
(8,T,Q)
be an arbitrary probability space and let
8}
be a family of random measures on
E'
denotes expectation w.r.t.
R .
S.
Then there exists
-13-
a unique probability measure
Mx T
R on
satisfying (1)
if
Lt.: (f)
e
e for every fixed
is T-measurable in
Proof.
c
The necessity of our condition follows by monotone convergence
from simple functions.
in
f E F
e for every
f
e
e
Lt.: (f)
is measurable
e
Fc
E
L~
is measurable in
Conversely, suppose that
Then
(L
j
t.f.) = E exp(-L t.t.: f.)
j
J (I J
J J
for any fixed
keN
Now i t follows by the entended Stone-Weierstrass
theorem (see Simmons (1963)) that any continuous function
on
g(x ,
l
,x )
k
which vanishes at infinity can be uniformly approximated by linear
combinations of the functions
Therefore
eXP(-L/jX j )
Eg(!;efl' ... '!;efk)
with
is measurable in
t l , ... ,t k
e
E
for any such
R+
g ,
and hence, by an easy application of Urysohn's lemma, it follows that
the probability
is measurable in
e
for any
Let us now define
P{[,e
c M}
V
is measurable.
t l , ... ,t k
R+
as the class of all sets
Then
V
V contains the class
M c M such that
is closed under proper differences
and monotone limits, and it contains
above that
E
M.
On the other hand, it was shown
C of all sets
-14-
and since
C is closed under finite intersections, it follows by Dynkin's
theorem that
f E
Fc
But by Lemma 1.1, the right-hand side of this relation equals
P{~e E
so
M}
is indeed measurable in
e for all
M, and
M EM, as required.
We shall now introduce some important point processes and at the
same time illustrate the use of L-transforms and mixing.
For any
s
E
5 , let
and note that the mapping
a point process on
w
PT
=
-1
E
s -+
N be defined by 0sB = lB(s) , B
°
whenever
5
E<\
B -+ M
is measurable
S
,
for the distribution of
intensity
is a random element in
E
B,
is
Hence
(5,B)
, , it is easily seen that
0,
Writing
has
= pT -1 = W and L-transform
E exp(-o,f)
Next suppose that
n
Z+
E
= Ee -f(T) = we -f
and that
T
f
l' ... ,T n
F .
E
are independent random
Then any point process
with common distribution w E M
n
which is distributed as
is called a sample process with intensity
L
j=l °T j
elements in
c;,
Os
nw.
5
By independence,
Ee
~
has L-transform
n
-~f
=
IT
E exp(-o, f)
j =1
Here we may mix with respect to
=
(we
-f n
)
n
=
r
-
~f
E
F .
v , regarded as a random variable,
to obtain a mixed sample process with intensity
L:e
f
j
=.E' ( we -f) V = ,I,If' ( we -f)
Ev· wand L-transform
f E F ,
-15-
where
S
[0,1] ,
E
is the (probability) generating function of
In the particular case when
1jJ (s)
and the L-transform of
(2)
Ee
-~f
~
v
= e -a(l-s)
S
A = aw = Et;.
[0,1] ,
E
This measure
~
For this purpose, let
Then the restrictions
S.A
J
Sl,5 , ...
E
A to
Since their sum
E~B =
(=
S.
L. E~.B
J
=
J
M is arbitrary
~1'~2'
on
f
E
S with L-transforms
F .
satisfies
= (L
L. (S.A)B
J
S.A)B
j
J
it follows by Lemma 1. 3 that
t,
-Sof
J =
JT
AB <
00
is a point process.
Ec
t,l'sZ' ...
-Cf
J
J
cxp ( - L(S . A) (l
.
J
B
E
B ,
J
seen from (3) and the independence of
-r,f = E If e
(':e·
j
A E
are totally bounded, and so
J
J
LjSj
F ,
B be a disjoint partition of
exp(-E;.f) = exp ( -(S.A) (1 - e -f ) )
J
E
A is totally bounded, hut
E
2
of
f
satisfying (2) when
there exist independent point processes
(3)
a, we get
becomes
we may also define point processes
S.
is Poissonian with mean
= exp(- a(l - we- f )) = exp(- 1..(1 - e- f ))
where we have put
unbounded.
v
J
Furthermore, it
that, for any
15
f ( F ,
4It
-16-
which shows that
satisfies (2) .
I;
justified, using the fact that
Any point process
intensity
A.
(2) that, for any
I;B , ... ,I;B
l
k
J
and
(S.A) (1 - e
I;
)
are non-negative.)
is a process of this type, it is easily seen from
kEN
and disjoint
B , ... ,B E
l
k
B , the random variables
are independent and Poissonian with means
Let us now consider a Poisson process
a
-f
J
satisfying (2) will be called a Poisson process with
I;
If
I;.f
(The formal calculations here are easily
E R+ while
~
EM.
I;a
AB l , ... ,AB
with intensity
By Lemma 1.4, we may then consider
a
.
k
aA , where
as a random
variable and mix with respect to its distribution to obtain a mixed Poisson
process
I;
with L-transform
Ee -E;f = E exp ( -aA(l - e -f ) )
f
E
F ,
where
= Ee -at
Ht)
is the L-transform of
t
2:
0 ,
More generally, it is possible by Lemma 1.4
a.
to consider
A= n
directed by
n, possessing the L-transform
in (2) as a random measure, thus obtaining a Cox process
f
(4)
E
F .
k
Let us finally consider some fixed measure
where
k E Z
+
= Z
+
u {oo}
and
t. E S ,
J
j
~
k
l.l
E N ,
say
l.l
=
I
<s
j=l t j
Further suppose that
are independent and identically distributed R+-valued random
variables, say with
L =
S
~
.
Then it follows from Lemma 1.3 that
-17~ = 1:.S.<\
J J
is a random measure on
j
Ee -H =
=
S , and we get for
f
E exp(- 1: S.f(t.)) = IT E exp(-S.f(t.)) = IT ~
.
J
J
exp(I log ~
J
0
j
J
f(t.)) = exp(]..I log
J
j
~
0
0
j
J
F
E
f(t.)
J
f)
By Lemma 1.4, it is permissible to mn with respect to
]..I
n
when regarded
as a point process, thus obtaining
f
E
F .
Any random measure with this mixed distribution will be called a (j-compound
of
n.
In the particular case when
only with probabilities
a p-thinning of
n.
1 - P
and
S attains the values
p
respectively,
~
a
and
1
will be called
In this case, (5) takes the form
(6)
f
E
F .
Scrutinizing the arguments used in the above constructions, it is
seen that they do not depend on the nature of
S.
Thus we may e.g. proceed
as above to construct Poisson processes with arbitrary a-finite intensities
on any measurable space
NOTES.
(s,B) .
!Iere and below, our main source is Kallenberg (1973a).
However,
Lemma].4 is new and is essentially a strengthening of proposition 1.6.2
in Kerstan, Matthes and Mecke (1974).
The present method on constructing
-18-
Poisson and Cox processes is simpler and more general than methods in
the current literature.
PROBLEMS.
1.1.
ReB we mean a ring which is closed under non-decreasing
bounded limits. Show that, if 1 is a DC-semiring, then B is the
By a o-ring
smallest a-ring generated by
1.2.
Show that Lemma 1.1 remains true for any semiring
Let
1
B be an arbitrary DC-semiring.
c
LI I
J
Let
B with
Show that Lemma 1.4 remains
.
J
c replaced by the class of all functions of the form
for arbitrary k E N , t l ,
,t k E R+ and
,I k E 1
II'
1.4.
c
F
true with
L~=l
1
B.
a(I) =
1. 3.
In symbols: B = a(l) .
1
~
be a Cox process on
S directed by
n
Show that
~
has independent increments, in the sense that ~Bl' ... ,~Bk are
independent for any kEN and disjoint Bl , ... ,B k E B , iff
this is true for n
Prove the corresponding fact for compound
point processes with
2.
8 ~ O.
(Hint: use analytic continuation.)
ATOMIC PROPERTIES
Let
M
d
Every measure
denote the class of diffuse (or non-atomic) measures in
~ E
M may clearly be written in the form
k
(1)
~ = ~d +
L
j =1
b.o t
J
j
M.
-19-
Md , k
for some
~d E
distinct
t l , t 2 , ...
E
= Z+
Z+
E
U {~}
,
b l , b 2 , ...
Moreover,
11 E
N iff
11d
E
Md ,
k
measurable functions of
E
S
on
S with
Lemn~
81
~
82
~
...
2.1 are by no means unique,
8 , 8 , ...
1
2
to be taken
and the corresponding atom positions
assume the basic probability space
e~lal
(rl,A,P)
8j .
T , T2' ...
l
(We shall always
to be rich enough to support
Though the decomposition
~
(2)
M such that
a.s. (which holds automatically when
~S < ~
to be chosen in random order within sets of
any such randomization.)
E
However, when considering random
is compact), we may require the atom sizes
in order
11
decomposition.
11
not even apart from the order of terms.
~
and
Z+
The measurable functions occurring in
measures
measu~able
There exists a decomposition (1) of' every
the quantities
(O,~)
and
""d -- 0
When considering random measures, we often need a
LEMMA 2.1.
=
S , and this decomposition is clearly unique apart
from the order of terms.
~e
E R~
= ~d
v
+
L
j =1
8·0
J T
.
J
is still not unique unless all the atom sizes are different with probability
1 , we obtain in this way a unique joint distribution of the random elements
occurring in (2).
Par the proof of Lemma 2.1, we shall need an auxiliary result, the
formulation and proof of which requires some further terminology and notation. _
-20-
Given any set
B E B , we shall say that
par' ti tions of
B
{B .} c B is a null-ar~y of
nJ
for fixed n E N form a finite disjoint
if the
B.
nJ
partition of B and i f max. 1 B 01
J nJ
in any fixed metrization p of S
P.)
independent of the choice of
*
pEN
£
and
pi
p* B
:::
£
p'B
£
LEMMA 2.2.
Let
(Note that the last condition is
p E M and
For any
:::
pB -
>
£
0 , we define
L
lJ{s}l{lJ{s}~d (ll)
£
lim
BE B ,
I
sEB
II EM,
{p{shd (p)
L
sEB
and
> 0
null-array of partitions of
B.
n-+<x> J
nJ
For sufficiently large
B E
0
~e:}
o
J
B , and let
{B o} c B
nJ
be a
(p) ::: p * B
e:
n EN, all ll-atoms in
will lie in different partitioning sets
L I {pB
BEB •
Then
L I {lJB
o
PROOF.
denotes the diameter
M by
E
£
I. I
where
-+ 0 ,
.~e:}
nJ
(ll)
B of size
~
£
Bnj , and we get
~
p*B
e:
Thus it remains to prove that
(2)
If (2) were false, there would exist some subsequence
numbers
jn'
n E N' , such that
N'
c
N and some
-21n
(3)
n
Choosing arbitrary
some further subsequence
n
lB. I
nJ n
Nil , and sin¢e
E
every open set
G
Nil
-+
c
E
tradicts the definition of
.
N' , there exists by compactness
and some
N'
s
B such that
E
0 , it follows from (3) that
B containing
E
N'
E
But then
s
ll'{S}
ll'G
£
e:
~
E
sn
~
-+
E
s ,
for
, which con-
Hence (2) must be true, and the lemma
II £'
is proved.
PROOF OF LEMMA 2.1.
Define the set function
= II *£ B
(4)
~
BE B
= ~(ll)
E >
by
0 .
By the Caratheodory extension theorem, (4) defines for each
unique measure
~
on
~
S x R: ' and it is easily seen that
atomic with all its atoms of unit size.
Lemmas 1.1 and 2.2 that
M a
is purely
Furthermore, i t follows from
is a measurable function of
~
II E
Suppose
II
k
~ , i.e. that
that the lemma is true for
functions
k E Z
+
and
cr , °2'
l
E
~
= I
0
for some measurable
j =1 cr j
S x R'
Then (1) holds with
+
(t . , b. ) , and since t. and b. are measurable functions of the
J
J
J
J
pair (t. , b. ) , the measurability in (1) will follow by Lemma 1.3. We
J
J
may therefore assume from now on that lld = 0 and b l :: b 2 - 1
cr.
J
-
Let
that
N'
c
p
s1' s2'
(s, sn)
...
= p (t,
N be such that
be an arbitrary dense sequence in
sn) ,
sn
-+
n
s
N , implies
E
n
E
s
=
t
S
and note
(To see this, let
N' , and note that then
-22p(S,t)
peS, s n )
5
=2
pes,
n t)
+
pes, s n )
We may therefore introduce a linear ordering of
whenever, for some
pes, s.)
j
J
= 1,
For any fixed disjoint partition
the atom positions
of
t.
J
We have to show that
S by writing
~
Bl , B2 , '"
E
s
<
t
B of S, we now order
, first according to their occurrance in
Bn , according to their linear order.
t l , t , ...
2
are measurable when ordered in
this way, and by induction it suffices to consider
{t l
N' .)
€
... ,k - 1
Bl , B2 , ... , and then, within each
that the set
n
kEN,
pet, s.)
=
J
0
+
B}
€
t l ' i.e. to prove
~ E
is measurable for every fixed
B.
Now this
set is clearly expressible by means of countable set operations in terms
c
(c n B.1 n {s: p (s, s. ) < r}) =
with C = B or B
of the sets
{p
and with arbitrary
i,j
E
N
,k
k}
J
E
Z+
and rational
r
Since
> 0
the latter sets are measurable, this completes the proof.
We shall now establish a simple relationship between the one-dimensional
distributions of a random measure
*
~£'
point processes
o<
a < b 5
00
THEOREM 2.3.
{B .}
nJ
o
<
c
£
>
~
and the intensities of the corresponding
0 , or more generally of
s;
Let
00
* -
* '
~b
•
~
be a random measure on
B be a null-array of partitions of
a < b
*
~[a,b) = ~a
Then
S, let B E B and let
B.
Further suppose that
-23-
*
E~[a,b)B
(5 )
holds provided Et;'B
a
<
= lim L P{~B
n-+-<o j
. e [a,b)}
nJ
, while in general
co
1iminf? P{t;Bnj e [a,b)} .
(6)
n-+-<o
PROOF.
J
By taking differences in Lemma 2.2, we get
(7)
lim? l{t;B .e[a b)}
n-+-<o J
nJ
'
and hence, by Fatou's lemma,
1iminf E
n-+-<o
proving (6).
= 1iminf L P{~B
? l{~BnJ.era ' b)}
n~
J
j
*
Since (6) implies (5) in the case
to prove (5) in the case when
*
and
E~Ia,b)B
.
E
[a,b)} ,
=
co
nJ
E~[a,b)
,
it remains
are both finite.
E~'B
a
But
then (5) follows from (7) by dominated convergence, since we have
I} u
{~'B
.
a nJ
~ a}
,
and hence
*
~ jL ~I a, b)BnJ.
\
+ L
j
a
-1
~*
B
~'Bh··
a J - ~[a,b)
We next give a simple criterion ensuring
size
~ £.
Given any covering class
C
c
t;
+
a- 1n B
sa
to have no atoms of
B and any
a > 0
we shall
-24~
say that
a-regu~ar
is
w.r.t.
C, if there exists for every fixed
C
E:
C some array {C .} cC of finite coverings of C (one for each
nJ
n
E:
N ), such that
lim L
P{~C . ~
.
nJ
(8)
a}
=0
.
n+oo J
THEOREM 2.4.
cZass and
~
Let
~et
be a random measure on
~*
Then
a > 0
= 0 a.s. if
a
The converse is aZso true provided
PROOF.
and let
~
Suppose that
{C .}
nJ
c
C
a}
S
C c B be a covering
~et
S,
~
is a-reguZar w.r.t.
C is a DC-semiring and
C, let C
is a-regular w.r.t.
be an array of coverings of
E:
E~ E:
C.
M.
C be arbitrary
C such that (8) holds.
Then
p{max
s C
~{s} ~
~C nJ.
P{max
.
J
*
~a
=0
of partitions of
~
a}
S
*C
a
=0
M and
~a
~
a.s.
Since
I.
J
C
P{~C . ~ a}
nJ
,
is covering, this
a.s.
Conversely, let
a DC-semiring.
{~C .
=P U
.
nJ
J
and it follows from (8) that
implies
a}
~
E~ E:
For every C
E:
C
*
=0
a.s. , and suppose that
we may then choose a nUll-array
C , and we get by Theorem 2.3
lim
P{~C . ~ a} + E~ * C = 0 ,
I.
nJ
n+oo J
proving a-regularity of
~
w.r.t.
a
C.
C
is
{C .}
nJ
c
C
-25It is often useful to replace the global regularity condition (8)
by a local one.
if
C is a DC-semiring and
For a first step, note that if
A E M is arbitrary, then (8) is implied by
SUP{A~ P{~C ~
lim
(9)
£+0
a}: C
E C
We shall show that the uniform convergence in (9) may essentially be replaced
by pointwise convergence.
THEOREM 2.5.
a >
Then
O.
(10)
~
Let
*
~a
lim sUP{Ai
£+0
be a
whepe
lim
£+0
P{~I ~
I
sup{~~ P{~C ~
a}: I
B
c
I
E
sET
S E
S ,
S
S ,
and also if
3
a}: C
SEC
C
E
P{~
Suppose that
I
*
a
~
a}
>
a ,
and let
we may assume
refinements.
I
c
E
B be a DC-semiring.
E
I
such that
{I.}
c
I
of partitions of
nJ
nJ
Then
a
>
and it follows that
a
>
O}
>
a
I , and
to be nested in the sense that it proceeds by successive
{I.}
Pis * I
Pis * I
I
is covering, we may choose some
We may further choose some null-array
(12)
M and let
A E
C is the class of closed or the class of open B-sets.
PROOF.
Since
S , let
measupe on
= 0 a.s. if
fop some DC-semiping
(11)
~ndom
O}
=
P U {~ * II'
j
a
J
>
O} ~
Ij
P{~ * II'
a J
> a}
,
-26(13)
max
j
provided
1
*
xr-pis II'
lj
a J
> a} ~
a
is interpreted as
%
from (12) if
AI
=
a , while if AI
In fact, this follows trivially
>
0 , insertion of the converse of
(13) into (12) yields the contradiction
P{s * I
a
By (13) there exists an index
such that
jl
1
*
All P{sall > a}
~
a} .
>
I 1 = II'
J
satisfies
l
1
*
rrP{sa l > O}
and proceeding inductively, we may construct a sequence
in
I
lIn' ~ 0
with
{I}
12
J
•• ,
n EN.
has a non-empty intersection
n
J
and such that
1
- - P{O
AI
n
n
Now
11
that (10) is violated at
{s}, and the last relation shows
s.
Let us now introduce the class
I = {B
where
aB
=
C
Bn B
E
B: AaB =
a} ,
denotes the boundary of
B
If we can show that
is a DC-ring, it follows that we may apply the first assertion to
Now the ring property of
a(B u C)
To see that
fixed
t
E
I
c
aB u ac
I
I.
follows from the elementary relations
a(B n C)
c
aB u ac
has the DC-property, it suffices to show that, for any
S , the ball
I
-27-
S(t,r)
belongs to
that
I
S(t,r)
compact.
< £
E
S: p(s,t)
for arbitrarily small
E
as(t,r)
c
is
E
since
< £ ,
AaS(t,r)
=
0
and hence
= r}
s: p(s,t)
S
is locally
,
S(t,r)
E
for all but countably
I
•
we have
I
1
IT
p{O
In the case of open sets, let
£
I
G :J I
by an open bounded set
AG
~
rAJ
P{t;I
>
AI
=
~ a}
which proves that (11) implies (10) when
I
For a proof, note first
Next verify that
With the above choice of
E
r}
<
r > 0
B for small r , say for r
and conclude that
many r
{s
=
I
I
E
I ,
E
0 be arbitrary and approximate each
such that
a}
AI
2l~I P{~I ~
a}
IGI
<
2111 v
,
=0
Then
1
IGP{~G ~ a} ~
and again (11) implies (10).
{
I , and hence
C is the class of closed sets.
AI > 0
~
AI
AI
>
AI
=0
0 ,
,
£
and
...28From Theorem 2.4 it follows in particular that, given any covering
class
C, a random measure
C for every a
w.r.t.
C
is regular w.r.t.
>
~
0
~
is a.s. diffuse provided
is a-regular
In this case we shall say, simply, that
~
Other diffuseness criteria are implicit in Theorem
2.5.
Another case of particular interest is that of point processes.
is a point process, we write
~
*
~
with
a.s.
a
taking
=
1
a
=
=
~
* and say that
1
=0
~i
Since for point processes
If
is simple if
a.s., (5) holds generally
Furthermore, we obtain convenient simplicity criteria by
in Theorems 2.4 and 2.5.
2
We conclude this section with a simple but useful lemma.
LEMMA 2.6;
Let
~
be a point process on
semiring.
Then
~
is simple iff
P{~I
(14)
PROOF.
>
I}
~ P{~ * I
>
S
I}
I ~ I
The necessity of (14) is obvious.
I
and let
c
B be a DC-
.
Conversely, suppose that (14)
holds, and note that then
(15)
0
Letting
of
~ P{~I > ~
I
E
I
* I = I} =
P{~I >
I} -
P{~
*I
>
I}
be fixed and choosing a null-array
I , we obtain by (15) and Lemma 2.2
~
0
I
{I.} c I
nJ
E
I .
of partitions
-29P{sI > s * I}
s P U {s*I .
j
nJ
*
~)I
proving that
(s
- s*
=0
~
that
NOTES.
~* I
P U {sI . >
j
nJ
=
> I}
.} = P U {sI . > s * I . > I}
nJ
j
nJ
nJ
=
p{I
j
=0
a.s.
*
1
>
.>0
nJ
Since I is covering, it follows
{s I
a.s., as asserted.
In the point process case, Theorem 2.3 with
a
=1
and
b
=
00
is known as Korolyuk' s theorem, while the converse part of Theorem 2.4
=2
with a
(1969).
first
is known as Oobrushin's lemma, see Leadbetter (1972) and Belyaev
The present generalizations to random measures are new,
~tep
towards Theorem 2.4 was given in Kallenberg (1975)
ideas behind Theorem 2.5 are implicit in Kallenberg (1973d).
while the
See also
Gikhman and Skorokhod (1969) for a version for random processes in
of the direct part of Theorem 2.4.
although a
0[0,1]
We further point out that our conditions
(10) and (11) generalize the notion of analytic: orderliness in Daley (1974),
while (9) generalizes his notion of uniform analytic orderliness.
Finally,
we mention that Lemma 2. I was given without proof in Kallenberg (l973b)
while Lemma 2.6 is taken from Kallenberg (1973a).
PROBLEMS.
2.1.
The measurability of
of Lemma 2.1.
of
).l
-+
).l~
).l
-+).l * ,
Give a direct
a
>
0
a
a
>
0 , is an immediate consequence
argument, proving the measurability
and conclude that
).l -+).ld
is measurable.
(All these facts are of course contained in Lemma 2.1.)
2.2.
Prove that the events
*
{t, = ~}
{s
are measurable.
= ~d}
(Hint:
and (in the point process case)
considE~r
the differences.)
~
-302.3.
Prove that the notion of null-array of partitions and the conditions
(10) and (11) are independent of the choice of metric
2.4.
Let
s
be a Cox process directed by
iff
n
is diffuse.
n
(p>O) , then
Prove also that, if
sand
n
s
s
is simple
is a p-thinning of
(Hint: consider
n .)
Show that Lemma 2.6 follows easily from Theorem 2.3 in the particular
E~2
case when
2.6.
Let
s
I
€
M.
be a random measure on
and let
2.7.
Show that
are simultaneously simple.
first the case of non-random
2.5.
n.
p •
a > O.
S, let
~*
Prove that
a
=0
1
c
B be a DC-semiring
a.s. iff
P{sI
~
a}
~
~
P{s'I
a
a} ,
1 .
€
Show by an example that the decomposition in Lemma 2.1 is not unique
in general, not even apart from the order of terms.
e.g.
v
S = [0,1] ,
=2
Bl
= B2 = 1
(Hint: take
, and choose
(T l , T2)
in two different ways.)
2.8.
Show that the converse of Theorem 2.4 may be false when
F~ ~
M.
(Hint: consider a suitable mixture w.r.t.
simple point processes on
3.
n E N of the non-random
n
R with atoms at j2j EN.)
UNIQUENESS
The uniqueness results of this section will playa basic role in
the subsequent discussion of convergence in distribution.
Let us start
with a theorem which applies to arbitrary random measures.
and
V
=
(B
l
,
...
,B
k)
€
Bk , kEN , we abbreviate
(llB
l,
For any
p
E
M
-31d
Write
for equality in distribution, i.e.
THEOREM 3.1.
Let
determined by
be a random measure on
E;
PCE;f)-l,
further determined by
f
E
d
~,
S.
P~-
iff
Tl
Then
E
= Pn -1
.
is uniquely
PE;-l
Fc ' and even by L~(f) , f
p(~V)-l , V
1
Fc
It is
r k , kEN, for any fixed DC-semiring
E
reB.
PROOF.
Let
rk
k
V
E
and
E;
E
n be two random measures on
N , for some fixed DC-semiring
the class of all sets
V contains
PE;-lM
M E M such that
J
c
=
S satisfying
B .
E;V
d
=
nV ,
Letting V denote
Pn -1 M , it is seen that
M and is closed under proper differences and monotone limits.
V contains by assumption the class
Furthermore,
C of all sets of the
form
k
f>
N
t l , ••. , t
k
E
R+
Since this class is closed under finite intersE!ctions, it follows by Dynkin' s
V:J o(C) .
theorem that
desired conclusion
But
o(C) = M by Lemma 1.1, and we reach the
, d
E; = n .
A similar argument shows that
all distributions
P (t;f l ' ... , E;f k )
p~-l is de!termined by the class of
-1
kEN,
f l' ... ,f k E Fc
But by the uniqueness theorem for mUlti-dimensional L-transforms,
,~fk)
L f
f
f;, l'H.,f;, k
where
f
-1
(t l' ... , t k )
= \.t.f.
L
J J
assertion.
J
is in turn determined by
Since
f E F , this completes the proof of the first
c
-32-
Note in particular that the L-transforms which were calculated in
Section 1 can be used to define the corresponding basic point processes.
COROLLARY 3.2.
n be a random measure on S and let t;,
-1
is determined by pt;,-l
n • Then Pn
Let
process directed by
is also true when
PB- 1 is known and such that
n "' provided
PROOF.
f
n is a point process on
Suppose that
in
1 - e
where
Ilfll
-f
t;,
B~
o.
is a Cox process directed by
, we obtain from
This statement
is a B-compound of
t;,
S and
be a Cox
(1.4) for any fixed
Solving for
n
f
E
F
c
(1)
=
sup f(s)
Now
L
nf
is analytic in the right half-plane,
and so it is completely determined by its values on any finite interval.
Hence by (1),
is determined by
pt;,-l ,
arbitrary, the assertion follows from Theorem 3.1.
and since
f
Fc
E
was
A similar argument
applies to the case of compound point processes, except that (1) is then
replaced by
o~
where
cP
-1
is the unique inverse of
cP
t <
-I If 11-1
on the interval
log P{B
=
O} ,
(p{ 8 = a} , 1]
Stronger results than in Theorem 3.1 are attainable if we confine
ourselves to simple point processes or diffuse random measures.
.
-33-
THEOREM 3.3.
that
~
~
Let
is simp le.
a DC-semiring.
~
U c B is a DC-ring and
= O} = P{nU = O}
d
=n
-1
M = Pn
B
provided in addition
I}
>
suppose that (2) holds.
p~
c
UE U,
~
P{nI > I}
I E I
.
The conditions (2) and (3) are clearly necessary.
satisfying
I
= n* iff
P{~I
(3)
S" and suppose
d
~
P{~U
and we have even
be two point proeesses on
n
Further suppose that
Then
(2)
PROOF.
and
V of all sets
As before, the class
-1
Conversely,
MEN
M is closed under proper differences and monotone
limits, and it contains N
By assumption, it: also contains the class
{~E
of all sets of the form
N:
= O},
~U
U
E
U.
C
Now it is easily seen
that
{~U
and since
= O}
n {~V
= O} = {~CU
V) = O}
u,V
E
U,
U is closed under finite unions, i t follows that
under finite intersections.
from Lemma 2.2 with
w. r. t.
u
a (C)
e:
=I
so we get
Hence, by Dynkin's theorem,
it is seen that the: mapping
~
= t;, * d= n*
C is closed
V:::> aCC).
~ -+ ~
*
But
is measurable
, proving the first assertion.
If in
addition (3) holds, then
Pin * I > I}
= P{~I >
and it follows by Lemma 2.6 that
I}
~
n
P{nI
>
= n* d= t;,
I}
•
I
E
I ,
This completes the proof.
-34-
Note that a variety of related results may be obtained by using the
simplicity criteria of Section 2 rather than condition (3).
A corresponding
remark applies to the next theorem.
THEOREM 3.4. Let
on
U
~
and
S, and suppose that
c
B is a DC-ping and
n be
ape fixed peal numbeps with
point ppocesses (pandom measupes)
is simple (diffuse).
~
C
~o
c
Fupthep suppose that
B a coveping class, and that
Then
0 < s < t.
~ g n iff
sand
n
t
is simple
(a.s. diffuse) and
UE U,
(4)
and also iff (4) holds and in addition
_ Ee -snC
Ee -s~C <
(5)
E~
In the case
C
E
C.
EM, we may peplace (5) by
E~C 2':
(6 )
EnC
C
E
C.
It is interesting to observe that (2) is a limiting case of (4) obtained
by letting
PROOF.
t
~
00
•
Let us first consider the point process case, put
p = 1 - e
-t
~
and let
~
and
n be p-thinnings of
~
and
n respectively.
we get
B
E
B
By (1.6)
-35-
and letting
(8)
x
-+-
~
P{~B =
(X)
,
we obtain by monotone convergence
O} = E exp ( ~B log(l - p) ) = Ee -·t~B
By (4), (8) and the corresponding relation for
P{~U
=
BE B .
n , we obtain
O} = Ee-t~U = Ee- tnU = P{nU = O}
and it follows from Theorem 3.3 that
~*
t;,
d
~*
=n
UE U ,
Now
and
are
simultaneously simple and correspondingly for n and n (cf. Problem
~ d ~
~ d ~*
2.4), so we get ~ = n
in general, and if n is simple even t;, = n .
Thus the first assertion follows by Corollary
~;.
2.
To prove the second assertion, we may assume that
(5) holds.
<
t;,
d
~*
=n
and that
Choosing
in (7) and in the corresponding relation for
s
~
n , which is possible since
t , we obtain from (5)
~*
Ee-xn C
~
= Ee-x~C
~
~ Ee-xnC
C
E
C ,
E
C.
or equivalently
C
But here the random variable within brackets is non-negative, so it must
in fact be a.s.
0, and it follows that
~
nC
=
~*
n C a.s.,
C
E
C
since
-36~
x
>
O.
Writing this result in the form
and using the fact that
~
d
~ =
~
n , as required.
(n
~*
n )C
=a
n =n
C is covering, we obtain
a.s.,
C
€
C,
a. s., so again
The last assertion follows by a similar argument,
~
based on the relations
E~
= pE~
and
En
= pEn,
which may e.g. be obtained
from (7) by differentiation.
The random measure case may be proved from Theorem 3.3 by a similar
~
argument where we consider Cox processes
and
tn
respectively.
and
~
n
directed by
Alternatively, we may choose any fixed
t~
x
>
t
~
let
and
~
n be Cox processes directed by
x~
and
xn
respectively,
and apply the point process case of the present theorem.
As an application, we prove the following partial strengthening of
Corollary 3.2.
p
=
>
O}
C
E
B with
some
by
p~-l
PROOF.
(9)
be a S-compound of n and suppose that
a . Further suppose that
P{S
>
~
Let
COROLLARY 3.5.
is known to be simple for
Cn
~ a . Then Pn -1 and PS-l are uniquely determined
nC
and p
By (1.5),
Ee-t~B
and letting
t
7
=E
exp(nB log ~(t))
~
we obtain by monotone convergence
,
P{~B = a}
B
E
= E exp(nB 10g(1 - p))
B
B
E
B .
-37Considering sets
P(Cn)-l
B
=
B contained in
C , it follows by Theorem 3.4 that
is uniquely determined by
PI;-l.
Let us now write (9) for
C in the form
LI;C
(10)
Since
= LnC
0
(-log
nC ~ 0 by assumption, the function
so it must have a unique inverse
<p)
•
L (~
n..
is strictly increasing,
on its ,range
(P{nC
=
Ol, 1] ,
and we obtain from (10)
proving that
<p
is determined by
PI;-l.
We may now apply Corollary
3.2 to complete the proof.
NOTES.
In Theorem 3.1, the first assertion is due to Prokhorov (1961),
(see also von Waldenfels (1968) and Harris (1971)), while the second assertion
is essentially a standard result for random prbcesses.
proved in the particular case of Poisson processes by
Theorem 3.3 was first
R~nyi
(1967), and then
in the general case, independently by Mt5nch (1971) (in a weaker formulation)
and Kallenberg (1973a).
As for Theorem 3.4, the first assertion was proved
for diffuse random measures (again in a weaker formulation) by Mt5nch (1970),
and independently by Ka1lenberg (1973e) and Grande1l (1973).
The point
process case of that theorem is new as are the last two assertions.
Finally,
Corollary 3.2 is due in the thinning case to MElcke (1968) and in the case of
Cox processes to Kummer and Matthes (1970), while the assertions for compound
processes in Corollaries 3.2 and 3.5 are new.
-38-
PROBLEMS.
3.1.
Show that, for any random measure
L~(f)
determined by
IT'
3.2.
~,the
distribution of
for all linear combinations
l E I , for any fixed DC-semiring
f
~
is
of indicators
1.
Theorem 3.1 clearly extends to arbitrary semirings generating
B.
Prove that the first assertion in Theorem 3.3 extends to arbitrary
rings
U generating
B.
(Hint: Show that the class of sets
U
satisfying (1) is closed under monotone bounded limits, and hence,
by the classical monotone class theorem in
B = a(U) .
Then apply Theorem 3.3 with
Lo~ve
(1963), must contain
U = B.)
Even the second
part of Theorem 3.3 admits a similar strengthening, cf. Kerstan,
Matthes and Mecke (1974), page 43.
3.3.
Prove an alternative to the second part of Theorem 3.4 by applying
the same method involving Cox processes to the second part of Theorem
3.3.
3.4.
Prove that the distribution of an arbitrary point process ~ is
not even determined by P(E,;B)-l , B E B
(Hint: consider two
2
d
such that a ~ 13 , and
random vectors a and 13 on {O,1,2}
still
3.5.
a l ~ 13
a
1
2
~ 13 2 and
al + a
Show that the ring property of
~ 13
2 - 1 + 13 2
.)
U in Theorem 3.3 is essential.
(llint: we may e.g. introduce a suitable dependence in a stationary
renewal process.)
Cf. Kerstan, Matthes and Mecke (1974), page
213, for a stronger result and further references.
3.6.
Prove that, for an arbitrary random measure
E~ determines
p~-l
iff
E~
=
a .
~,the
intensity
-39-
3.7.
Show that
as
t-l-O,
B
E
B .
This is
interesting since, for simple point processes and diffuse random
measures ~, p~-l is determined by the left-hand sides for arbitrarily
small
3.8.
t > 0 , and still it is not determined by the limit.
Assume that
function f
S is countable. Show that there exists some fixed
E F such that
p~-l is determined by P(~f)-l for
every point process
numbers
f(s) ,
s
on
~
E
S with
~S <:
00
a.s.
(Hint: let the
S , be rationally independent and bounded
above and below by positive constants.)
3.9.
C in Theorem 3.4 is a DC-semiring, then
Show that, if the class
~
may be allowed to have atoms of fixed size and location.
(Cf.
the proof of Theorem 7.8 below.)
4.
CONVERGENCE
In view of Theorem 3.1, it appears natural to say that the random
measures
/;n'
n EN, converge in distribution to
~
(wri tten
whenever
(1)
~nV ~ ~V
for some fixed DC-semiring
kEN ,
I, and this is also the mode of convergence
used in much classical work (with
I
(1) requires too much in general since
does not necessarily imply
oT ~
n
0
T
the class of real intervals).
T
d
n
-l-
T
However,
for random elements in
s
in the sense of (1), (cf. the trouble
with pointwise convergence of distribution functions).
Moreover, the above
-40mode of convergence depends strongly on the choice of
I
As will be
seen below, we can avoid these disadvantages by assuming that
sal
0
=
a.s.,
do exist.
B
B~
satisfies
with this property
= 0 a.s.}
~aB
~
on
S.
Proceeding as in the proof of Theorem 2.5, we need only show that
~{s E
for any fixed
s
= {B E B:
I
is a DC-ring for every random measure
s
(2)
let
We prove that OC-semirings
Let us write
LEMMA 4.1.
PROOF.
lEI.
I
>
0
t
E
S
S: p(s,t)
= r} = 0 a.s.
and arbitrarily small
r
>
O.
For this purpose,
be such that
Set, s+)
=
{s E S: p(s,t) ::; dEB,
and define the random process
X(r)
X in
= ~S(t,
O[O,s]
r+)
r
E
by
[O,s] .
Then
~{s
E S: p(s,t) = r} = X(r) - X(r-)
and since we can have
r
E
(D,s]
P{X(r)
~
X(r-)}
>
0
r
E
(O,s] ,
for at most countably many
(cf. Billingsley (1968), page 124), it follows that (2) holds
almost everywhere on
[O,s]
This completes the proof.
-41-
To be able to apply the powerful theory of weak convergence of probability measures, we shall now introduce a metrizable topology in the
measure space such that (1) subject to the above restriction on
I
is
equivalent to convergence in distribution in the sense of this theory.
For any
to
)J
11, ]11' 11 2 ,
and write
~
•••
v
~ ~
n
in
N , we shall say that
M or
whenever
induced vague topology in
11
n
~)Jf
f
for all
f
l1
E
n
tends vaguely
Fc
The so
M or N is known to be metrizable and separable
(see Bauer (1972); it is indeed even Polish, see Lemma 10.1 below).
It is
often useful to relate the notion of vague convergence to the better known
notion of weak convergence.
we say that
l1
For any
tends weakly to
n
11, 11
Il
w
n
-+
iff
11
11
v
n
~)J
)J S
and
n
E
and write
lJ
f
E
F
)JS
<
00
for every bounded continuous function
that
1 , )J2' ...
~
M with
whenever
l1S <
00
11 f ~ l1f
n
The reader should verify
The next lemma shows that the random elements in the topological
Borel spaces
M and
N coincide with our random measures and point processes
respectively.
LEMMA 4.2.
M and N coincide with the a-algebras generated by the vague
topologies in
PROOF.
M and N respectively.
The vague topology in
M (or
topology induced by the mappings
)J
~
N) is by definition the product
pf
f
E
Fc , so the class C
kEN, is a topological base.
separable, every vaguely open set is contained in
Since
a(C)
,
M (N)
and so
is
a(C)
-42-
is the a-algebra generated by the topology.
aCe)
=
M (or
N) be Lemma 1.1.
On the other hand we have
This completes the proof.
Unless otherwise stated, convergence in distribution of random measures
d
an d point processes (d enote d
~,or
~)
~
.
sometlmes
. 1 h ence f ort h mean
wll
weak convergence of the corresponding distributions with respect to the
~n
vague topology, i.e.
i ~
Ef(~n) ~ Ef(~) for every bounded and
iff
vaguely continuous functional
f: M ~ R+
N ~ R+
or
respectively.
Con-
tinui ty in distribution holds for a large class of linear functionals.
Let
Of
denote the set of discontinuity points of the function
Let us further denote the mappings
~ ~ ~f
and
~ ~ ~B
by
w
f
f.
and
wB respectively.
LEMMA 4.3.
~
d
~ ~.
n
~, ~l' ~2'
Let
~n f
Then
i ~f
PROOF.
VE
Write
G E B with
g
E
F
c
k
B~,
G
C
E
E
0 a.s.
such that
f: S
~ R+ with
Furthermore~
~
C
B for the support of f
~ nV
i
~V for
and choose some open set
By Urysohn's lemma, there exists some function
Assuming
such that
gh
~Df =
S
kEN.
continuous function
since
be random measures on
for every bounded function
compact support satisfying
every
...
h
E
~
v
n
~ ~
, we get for any bounded
F
Fc , which proves that
If we further assume that
-43-
)JD f
a ,
=
i t follows by Theorem 5.2 in Billingsley (1968), (which clearly
remains true for non-normalized measures), that
This proves that
(3)
The first assertion now follows by Theorem 5.1 in Billingsley (1968).
Specializing to indicators
f
= lB'
B
E
B , it is seen from (3) that
B
B.
E
Hence the second assertion follows by applying Theorem 5.1 in Billingsley
(1968) to the mapping
)J
+
)JV,
V
k
E B~,
kEN.
For point processes, convergence in distribution means the same thing
in
M and N.
LEMMA 4.4.
In fact,
N is a vaguely closed subset of M.
processes on
S
M.
Suppose that
we have
such
closed under convergence in distribution w.r.t. the
1.-S
vague topology on
PROOF.
B
The class of point
)J B + )JB
n
E
)Jl' )J2'
for any
B
E
N with
B with
)JaB
v
)In
+
)J
E
M
, so )JB
= 0
By Lemma 4.3
E
Z
+
for any
Applying Dynkin's theorem as in the proof of Lemma 1. 2, it
follows that
)JB
E
Z+
for all
B
E
B , and so
)J
E
N
This proves that
-44-
N is a vaguely closed subset of M. Now suppose that
are point processes on
~.
S
~l' ~2'
'"
converging in distribution to some random measure
By Theorem 2.1 in Billingsley (1968), we obtain
P{~ E N} ~
~
proving that
P{~n E N}
limsup
n-+oo
N a.s., i.e. that
E
~
=1
,
is a point process.
We shall now prove the basic fact that the classical notion of convergence introduced above is essentially equivalent to convergence in
distribution with respect to the vague topologies.
THEOREM 4.5.
I
c
B~
be a DC-semiring.
Then the foZlowing statements are equivalent.
d
(i)
~
(ii)
~ n f i ~f
(iii)
L~ (f) -+ L~ (f)
-+ ~ ,
n
f
E
n
~nV
(iv)
... be random measures on S and Zet
~, ~l' ~2'
Let
i ~V
F
c
f
E
Fc
kEN .
Two lemmas are needed for the proof.
LEMMA 4.6.
Le t
ia tight iff
rl' ~r , ...
2
~
{t.: B}
n
be random measures on
is tight for each
B
E
B •
S.
Then
{r}
. ~n
-45-
PROOF. We shall make use of the fact (cf. Bauer (1972)) that a set K
in
M or N is relatively compact iff
sup pB <
(4)
B
00
B.
E
PEK
{~n}
Let us first suppose that
for each
£
some compact set
> 0
P{~
(5)
For fixed
B
E
is tight.
K}
E
n
~
B we get by (4)
such that
K c M
1 -
By definition, there exists
n EN.
£
m = sup pB <
':0
,
and hence by (5)
pEK
P{~
n
proving tightness of
B
>
~ P{~
m}
{~nB}
the constants
~ £
n EN,
.
{~nBj
Conversely, suppose that
Gl , G2 , ...
4 K}
n
be open B-sets with
ml , m2 , ... > 0
Gk
is tight for each
+S
B
E
B.
, and choose for fixed
Let
£
> 0
such that
k , :n EN.
Define
K
=n
{p EM: p G
k
k
and note that
some
k
<::
~ ~}
,
K is relatively compact, since there exists for any
N with
Gk::J B , and therefore
sup llB
llEK
~
sup pG
llEK
k
~
m
k
<
00
•
B
E
B
-46~
Furthermore,
p{~
n
4 K}
P u {~ Gk
k n
=
U c
Let
~,n
k
>
mk }
s;
£
I
2-
k
2£
=
k
{~n}
proving tightness of
LEMMA 4.7.
L P{~nGk
> ~ } S
K
~l' ~2'
and
B be a DC-ring and let
be random measures on
be fixed.
> 0
t
let
S,
Further suppose that
~
d
4-
n
n
and that ei the 1"
liminf
(6)
P{~
n
U > £} S
P{~U ~ £}
U
U
E
£
(0,1) ,
E
or'
limsup E
(7)
exp(-t~
J}+<lo
Then
a. s.
U)
U
exp(-t~U)
Suppose that (6) holds and let
FEB
By Lemma 4.1 there exists for every
such that
C
~
F and
P{~C ~
such that
U
~
C and
P{~(U\C)
P{~U ~
Now
E
~
= 0 a.s. for every olosed Bet FEB with
nF
PROOF.
n
£} s
P{~C ~
£/2}
~
£/2
s;
£/2}
£/2}
£
s;
U.
E
~F
= 0 a.s.
be closed with
>
0
some closed set
We may next choose some
£/2.
Then
+ P{~(U\C) ~
£/2} s £ .
~ n C ~ nC by Lemma 4.3, so we obtain
P{nF > d
s;
P{nC > d
s;
liminf
n4-<lO
P{~
s;
liminf
n4-<lO
nU
>
~F =
d
P{~
C> d
n
S P{~U ~
d s:
£
,
0
CE B
n
U
E
U
-47and since
was arbitrary, it follows that
€
=0
I1F
a.s. as asserted.
A similar proof applies to the case when (7) holds.
PROOF OF THEOREM 4.5.
in Lemma 4.3.
The fact that (i) implies (ii) and (iv) was proved
Since (iii) follows trivially from (ii), i t remains to
prove that (iii) and (iv) both imply (i).
Let us first suppose that (iii) holds.
L~
(tf)
Then
L~(tf)
+
f E
n
Fc
so (ii) holds by the continuity theorem for L-transforms.
is tight, and so it follows by Prohorov's
{ ~}
and Lemma 4.6 it is seen that
"'n
Nil
This means that every sequence
such that
we obtain
E; f
n
~n
.i
some
.i
nf
(n
n
E
(n
Nil) ,
Fc ,we must have E;f d= nf,
f
E
E;
gn
, and since
N'
Nil) .
E:
f
F
€
f
c
E
N'
is relatively
{ ~}
theorem (Theorem 6.1 in Billingsley (1968)) that
compact.
But from (ii)
"'n
N contains a subsequence
c
From the implication (i) ==> (ii)
~ nf ~
, and since also
Fc
i;f ,
By Theorem 3.1, this implies
was arbitrary, we get
E;
n
.i
E;
(n
E
N)
by Theorem
2.3 in Billingsley (1968).
To prove that (iv) implies (i), the same argument goes through, except
for the point where we conclude from
(n
E
Nil) , V E I
provided
I
c
k
,
kEN.
~
"'n
d
+
n
(n
E
Nil)
d
that
E; V + nV
n
Now this step is permitted by Lemma 4.3
B , and we shall show that this relation is indeed true.
n
For this purpose, let
U be the ring of finite unions of I-sets, and
note that these unions can always be taken disjoint.
fact that addition is a continuous operation from
By (iv) and the
to
R
+
, we ohtain
-48-
~
d
n
U~
~U,
U c U , and so
liminf P{~ U >
n
n-+oo
£}
limsup P{~ U ~
$
n
n-+oo
Hence it follows by Lemma 4.7 that
FEB
with
E;F = 0
£}
nF
P{~U ~ £}
$
UE U
£
>
a .
a a.s. for every closed set
=
a.s., and in particular
n31 = 0
a.s.,
I EI ,
as desired.
We shall now improve Theorem 4.5 in the particular cases of convergence towards simple point processes and diffuse random measures.
any covering class
{~n}
c
B , and any a
of random measures on
every
each
C
C E C some array
0 , we shall say that the sequence
>
is a-regular w.r.t.
S
{Cmk }
c
Given
C if there exists for
C of finite coverings of
C
(one for
mEN) such that
lim limsup
(8)
n-+oo
JII+<X>
L P{t; nCmk
Note that, in the particular case when
reduces to a-regularity of
shall further say that
{E;n}
~ a} = 0 .
k
~l
= t;2 = ... = t; , this notion
E;, as defined in Section 2.
As before, we
is regular, if it is a-regular for every
a > 0
THEOREM 4.8.
that
~
Let
t;, E;l' E;2' ...
is simple.
a DC-semiring.
Then
be point processes on
Further suppose that
~ n ~ ~ iff
U c BE;
~s
S and suppose
a DC-ring and
I
c
BE;
-49-
lim
(i)
P{~
n
n-+oo
limsup
(ii)
U = O} =
P{~
n-+oo
{~B}
(iii)
Moreover..
n
t;,
i ~
n
n
I
I}
>
B E
U ,
E
I}
~ P{~I >
lEI ,
B .
follows from (i) and (ii) if
{~n}
1 imsup EE;, I
n
n-+oo
PROOF.
U
is tight for every
1 and from (i) alone if
(9)
= O}
P{~U
'/-s 2-regular w.r. t.
t;,
is 2-regular w.r.t.
EO <
~
1
or if
lEI .
00
The necessity of (i) - (Hi) follows by Lenunas 4.3 and 4.6.
Con-
versely, assuming (i) - (iii) • it follows by Lemma 4.6 that
{~n}
tight and hence relatively compact, so any sequence
must contain
a sub-sequence
N"
point process by Lemma 4.4.
B
E
~n
such that
i
some
n (n
N' c N
Nil)
E
Here
is
n is a
naB = 0 a. s. ,
By (i) and Lemma 4. 7 we get
U u 1 , and so by Lemma 4.3
t;, B
n
d
-+
(n
nB
E
Nil)
BEUuI.
Hence
(10)
r{nU
O} =
p{ nI > I} =
lim Pit;, U
n
nEN"
= O} = P{~U =
lim
nEN"
>
p{~
n
I
I}
and it follows by Theorem 3.3 that
this implies
~
O}
U
limsup Pit;, I
n-+oo
~
d
=n
n
>
Since
E
I}
N'
U ,
~
P{t;,I > l}
was arbitrary,
I
E:
1 ,
-50-
Now suppose that
Let
of
I E I
is 2-regular w.r.t.
~
be fixed, and let
{I.} c I
the number of sets
limsup
p{~
n+oo
n
I
>
I mk
r }
m
=2
a
for fixed
and
C . :: I .
nJ
If
nJ
r
is
m
mEN, we get by (ii)
limsup P U {~ I k
n+oo
k
n m
~
and that (ii) holds.
be an array of finite coverings
nJ
such that (2.8) holds with
I
I
> I}
~ limsup
n+oo
L P{~
k
I k > I}
n m
Hence by (2.8)
lim limsup P{~ I
n-+«>
n
>
{~nI}
.
m-+oo
proving tightness of
r } ~ lim L P{~I k
m
m-+oo k
m
> I}
=0
,
Therefore, (iii) follows from (ii) in this
case.
Next suppose that
Then tightness of
{~n}
contain a sub-sequence
is 2-regular w.r.t.
{ ~}
"'n
Nil
such that
more, it follows from (8) with
I
k
so
n
d
~ =
P{nI k
m
>
I}
= lim L lim
m+=
k nE Nil
is 2-regular w.r.t.
n , and again
some
naB
=0
~ ~ n*
particular (10) remains true and
lim
and that (i) holds.
follows as above, so any sequence
we may conclude from Lemma 4.7 that
m+""
I
a
=
2
P{, I k
n m
c
N must
(n E Nil)
B E Uu I
As before
so in
follows by Theorem 3.3.
and
>
a.s.,
n
N'
Cmk :: I mk E I
that
I} ~ lim limsup L P{~ I k
m-+oo
n-+«> k
n m
I , and hence simple by Theorem 2.4.
~ n ~, follows since N'
Further-
was arbitrary.
>
I}
=
0 ,
Therefore
-51-
By ~ebysev's inequality,
Finally suppose that (i) and (9) hold.
(9)
{~
implies tightness of
by Lemma 4.6.
If
as above that
~
~
d
n as
+
n
= n* and
d
~
for every
I}
n
n
n
+
I
I , so
E
through some
00
I +d nI
is tight
N" eN, then it follows
(n EN") , I E I , so by (9) and
Fatou's lemma (in the formulation for convergence in distribution, see
Billingsley (1968), Theorem 5.3),
E~I
=
En * I
En!
~
~
liminf
E~
nEN"
lienee
EnI
En * I
=
for all
I
E
we may conclude that
n
I
~
limsup
n+OO
I , and
E~
n
I
~ E~I
I
n must be simple.
E
I .
As before,
The proof is now complete.
From Theorem 3.4 we obtain the following analogeous result which
may be proved in the same way.
on
~, ~l' ~2' ...
Let
THEOREM 4.9.
~
and suppose that
S
sand
t
c
B~
-t~
(i)
lim Ee
n+oo
(ii)
liminf Ee
n+oo
(iii)
{~nC}
-s~
= Ee-t~U
is tight for every
~n
1. ~
w.r. t.
01'
if (9) holds with
1-
~n
~
Uc
B~
~s
iff
UE U ,
C
n >- Ee -s~C
Moreover,
_ +d
Then
suppose that
and that
0 < s < t
a covering class.
nU
Fu~ther
is simple (diffuse).
are fixed real numbers with
a DC-ring and C
C
be point processes (random measures)
C
C
E
E
C ,
C .
follows from (i) alone if
{~n} is 2-regular (Y'cgular')
C in place of
I
-52-
In many applications we need to consider convergence in distribution
with respect to the weak rather than the vague topology on
wishing to emphasize the underlying topology, we write
of
d
M.
wd
or
--+-
When
vd
--+-
instead
No new theory is needed for the case of the weak topology, since
-+
this case may easily be reduced to that of the vague topology:
THEOREM 4.10.
... be a.s. bounded random measures.
~, ~1' ~2'
Let
Then the following statements are equivalent.
(i)
~n ~ ~ ,
(ii)
~n ~
~
and
~
(iii)
~n ~
~
and
inf 1imsup P{~ B > d
n
BEB n-ro>
nS
d
-+ ~S
•
C
=
a
£
>
a
Note that (i) is trivially equivalent to
The point is that the latter condition may be replaced by the seemingly
much weaker conditions (ii) or (iii).
PROOF.
Since
~ ~ ~
iff
~ ~ ~
and
~
n
S -+
~S
, the weak topology is
metrizab1e, so the theory of convergence in distribution in metric spaces
applies to it.
in
Write
M(w)
and
M(v)
for the space of bounded measures
M endowed with the weak and vague topology respectively.
first suppose that (i) holds.
into
(1968).
M(v) x R
+
Since the mapping
~ -+ (~,
~S)
Let us
from
M(w)
is continuous, (ii) follows by Theorem 5.1 in Billingsley
-53-
Next suppose that (ii) holds, and let
{i; n S}
is tight, we may choose some
P{i; S
(11)
n
a
£,0
a
>
a
>
n EN.
> a} < 0
E(l - e
~
so by (ii),
limsup
n-+<x>
P{I;
n
B E B with
_I;B C
-a
) < £oe
,
and by Lemma 4.1 we may assume that
<'
Since
such that
By dominated convergence, there further exists some
(12)
be arbitrary.
=a
l;aB
a.s.
Using (11), we obtain
-I; nS
-1
c a-I; n S
C
I; B e
I; B > £ , I; S $ a] + 8 $ £ eaEI; BCe
n
n
n
n
C
I; B
-I; S
-I; B
-I; S
n
-1 e a E ( e n
-1 a (
n
en) + 8 ,
E
+ 8 = £ e E. e
1) e
E[£
-1
(12) and Lemma 4.3,
BC
>
£}
$
-I; B
-I; S
nn
£ -1 ea
lim E(e
- e
)
+
8
n-+<x>
= £-leaE(e-I;B
_ e-I;S)
+
0
= £-l e a E[e-I;B(l
0
$
8
C
_ e-i;B )]
+
8
C
$
Since
£-l e a E (l _ e-I;B )
+
+
8
0 was arbitrary, this proves (iii).
Let us finally suppose that (iii) holds.
some
28 .
=
Bk E
BI;
such that
Bk
n-+<x>
Sand
t
limsup P{l;n~
Choose for each
>
1
k- }
<
k-
1
n
EN.
kEN
+
0
-54-
Then
lim limsup P{~nB~
k-+-co
n+co
(13)
We now define the metric
> 8}
=0
£
0 .
>
in the topological product space
p
M(v)
x R
+
by
lll' ll2 ( M
where
is any metrization of
M(v)
Then
= ~ n BkC
I~ n Bk - ~ n Sl
n,k EN,
so (13) may be written
£
Furthermore, we have in
(~, ~Bk)
(15)
M(v) x
-+-
(~,
R+
~S)
and finally, since the mappings
II
M(v) -+- M(v) x R
+
II
(16)
at all points
d
(~n ' ~nBk) -+- (E;,
> 0 .
~Bk)
k -+- co ,
a.s. as
are continuous
-+- (ll, ll~)
with
as
lldB
k
n -+- co
=0
kEN .
Using Theorem 4.2 in Billingsley (1968), we may conclude from (14), (15)
and (16) that
(17)
-55Now the mapping
is continuous
~
M(v) x R
+
M(w) , and therefore
(i) follows from (17) by Theorem 5.1 in Billingsley (1968).
This completes
the proof.
NOTES.
The equivalence of (i) - (iii) in Theorem 4.5 was proved by Prohorov
(1961) for compact spaces and in general by von Waldenfels (1968), (see also
Harris (1971)). The remaining equivalence was given in Kallenberg (l973a).
This is also the source for Theorem 4.8, except for the assertion involving
(9) which was added by Kurtz (1973), while Theorem 4.9 improves and extends
a result by Kallenberg (1973e) and Grandel1 (1973).
Theorem 4.10 is new,
although it was used implicitly in Ka1lenberg (1973c) and (1974b).
We
finally refer to Jagers (1974) for Lemmas 4.2 and 4.4.
PROBLEMS.
4.1.
Let P, PI' P2 , ... E M and let I E Bp be a DC-semiring.
that p ~ p iff p I ~ pI, I E I .
n
Prove
n
p ~ p iff limsup p F :0; pF , FEB closed, and
n
n
liminf 1J nG ;:: :J.IG , G E B open. Note that one of these conditions
is not enough.
4.2.
Prove that
4.3.
F €: B be closed and
that the p-sets {pF < t}
Let
G E B open, and let t E R
Show
+
and {pG > t} are open in M and N
(Hint: use Problem 4.2.)
4.4.
Prove that, for bounded
and p S ~ pS
n
p, PI' P2'
...
EM,
Pn
w
~
P
iff
Pn
v
~
)J
-56-
4.5.
~'~l' ~2'
... be random measures on S and let Cl , C2 , ... E B~
o
wd
C t S. Prove that ~ ~ ~ iff Ck~n -+ Ck~ , kEN. More
k
vd
n
d
generally, ~ -+ ~ implies f~ ~ f~ for any bounded function
n
n
Let
with
f
E
F with bounded support satisfying
4.6.
Prove that, i f
'2'
" '1'
d ,
+
o
~
0
iff
then
T
l
n
n
4.7.
Prove Theorem 4.9.
4.8.
Extend Theorem 4.8 to the case when
W
f
=
0
a.s.
are random elements in
,
~1'
~2'
...
(S, B)
are general random
measures.
4.9.
d
Let ~'~1' ~2' ... be random measures on S such that ~n + ~
ft
.,.
and let 6: ~ 0 be fixed. Show that "'6:+
~
=
~
- 0 a.s. i f
- "'(e:,oo) {~n}
is a-regular w.r.t.
is also true whenever
E~ E
B~
for every
M
,
a > e: , and that the converse
(Cf. Kallenberg (1975).)
4.10. Show that, if the class C in Theorem 4.9 is a DC-semiring, then
~
may be allowed to have atoms of any fixed size and position.
(Cf. Problem 3.9.)
4.11. Show that the equivalence of (i) and (iii) in Theorem 4.10 may essentially
be restated as a tightness criterion for random measures subject to the
weak topology in M. Give also a direct proof of this criterion.
be Cox processes directed by n , n , ... respectively.
1
2
d
Show that ~ + some ~ iff nn .+ some n , and that in this case ~ is
n
Prove the corresponding result for pa Cox process directed by n
4.12. Let
~1' ~2'
.. .
d
.
thinnings with fixed
p > 0
•
-57-
4.13.
d
Let
n, n , n , ... be random measures on
l
2
suppose that the functions f, f l , f 2 , '"
wi th uniformly bounded support and satisfy
n{s
€
S: fn(sn) + f(s)
for some
sl' 52' ...
d
~fn + nf .
Show that in this case
S with ~ + nand
are uniformly bounded
€ F
+
s}
= 0
a.s.
(Cf. the proof of Theorem
8.1 below.)
5.
EXISTENCE
In the limit theorems of the preceding section, the limiting random
measure
~
was always assumed to exist.
We shall now show that, under
mild addi tional conditions, the existence of
hy compactness.
E;
will hold automatically
This provides a new approach to the general existence
problems in random measure theory.
We begin with the counterpart to Theorem 4.5.
LEMMA 5.1. Let
be a DC-ring.
E;l' E;2' •••
S
and let
Uc
B
Suppose that either
f
(i)
(ii)
be random measures on
d
~nV +
c,v
un: th
some
SV'
are such that
uk.y au .
€
F
c
or
k
€
N , where the random vectors
whenevey;'
U, Ul '
u2 '
..•
€
U
-58-
Then
l;
§. some
n
satisfying
l;
f
E
F
c
V E Uk ,
01'
kEN , respectiveLy.
PROOF.
In both cases,
{l;n}
is tight by Lemma 4.6, so by Prohorov's
theorem,
/; §. some l; as n
n
-+
/; f
n
we obtain
§. /;f
E
then
(n
Nt) ,
f
00
through some sequence
Nt
N
c
But
F
, by Theorem 4.5, so assuming (i).
c
I;f ~ I;f , and the assertion follows by Theorem 4.5.
E
In case of (ii), we may proceed as in the proof of Lemma 4.7 to conclude
1;3U
that
=
a
U
a.s.,
E
U
But then
I; V ~ I;V
n
Nt) ,
(n E
V
Uk ,
E
kEN, by Theorem 4.5, and the assertion follows as before.
Arguing in the same way, we can easily derive analogeous results
for convergence towards simple point processes and diffuse random measures.
For example,
LEMMA 5.2.
Let
be a DC-ring and
1.1'. r. t.
I
I
P{I; U
n
Further suppose that
Un -t aB
I;
(2)
Suppose that
{I;n}
Uc B
is Z-reguLar
and that
(1)
with
B a DC-semiring.
c
S, and Zet
be point processes on
1;1' I;Z'
=
O}
-+
~(Un) -+
some
~(U)
UE U .
1 whenever VI' Uz,
...
E
U and
Then there exists some simpZe point process
d
n
-+
I;
and
P{l;V
= a} = ~(U)
UE U .
l;
BE Uu I
on
S
-59-
We shall now use Lemma 5.1 to derive two important special cases
of a general existence theorem corresponding to the second part of Theorem
For notational convenience, we consider random vectors in
3.1.
random measures on the space
THEOREM 5.3.
on
Uc
satisfying
S
the distT'ibutions
(ii)
~V{l} + ~v{2}
with disjoint
PROOF.
V
and
Uk ,
E
= ~v{3} a . s . wheneve:'('
VI
and V2 ,
V
For fixed
U with Vn
E
~n = ~ ~v{j}os . ' where
Writing
Un
nJ
E
U , let
V , choose arbitrary
, and define the random measures
J
kEN, iff
p~-l
aT'e consistent,
V
The necessity of (i) - (iii) is obvious.
of nested partitions of
kEN,
Then theT'e exists some T'andom meaSUT'e
~V ~ 0 wheneveT' V, VI' V2 , ...
k
that (i) - (iii) hold.
j
B~
(i)
(iii)
kEN
U c B be a DC-T'ing and foT' any V E Uk,
~V be a T'andom vectoT' in R~
let
~
Let
{I, ... ,k} ,
8S
~l'
~2'
av .
Conversely, suppose
{U .} c U
nJ
Snj
on
E
S
be a null-array
for all
Vnj
, Vn2 ' ... ) ,
V = (U nI
for the ring generated by
...
-I-
nand
by
n EN.
n I' Un.2' ... , we obtain by (i)
V
and (ii)
kEN ,
-60-
and so
~
where
U' = U Un
d
V+
n
Now
~
kEN ,
V
U'
is a DC-ring in the subspace
n
follows by (iii) and Lemma 5.1 that
~ n ~ some ~'
V , so it
satisfying
kEN .
(3)
To see that (3) extends to
U n
V , let
mEN
and
VI' ... ,Um E
U n
V
be arbitrary, and repeat the above argument with a refined null-array
such that
U'
is replaced by some
U":::>
the existence of some random measure
~"V 9:- t; V
(4)
<,
~"
U,
U
{Ul'
,Vm} .
This yields
such that
kEN .
Comparing (3) and (4), we obtain
t; 'V d t;"V
and so
t;'
~
E;"
by Theorem 3.1.
kEN ,
Inserting this into (4) yields the desired
extension of (3).
To complete the proof, let
t;i,
t;2 .. ,
as
lJ.
be random measures
VI' U2 , ... E U with U~ + S , and let
t;' obtained as above with VI' U , ...
2
Then it follows by another application of Lemma 5.1 that
with the asserted properties.
t;k
~ some
t,
-61-
THEOREM 5.4.
k
R+ .
For any
V
k
B
E
kEN , let
,
~
Then there exists some random measure
k
V E B
,
d·
= ~V
~V
'
a. 8. whenever'
with disjoint
(iii)
d
E;B
+
n
and
B
1
° whenever
B ,
2
B , B , ...
1
2
The necessity is again obvious.
- (iii) hold.
is a ring.
and put
satisfying
S
PE;~l are aonsist;ent,
the distributions
(i i)
PROOF.
on
iff
kEN,
(i)
be a random veator in
U = {B
Define
and suppose that
=
{s
8
E
S: p(s,t)
°
>
(5)
for infinitely many
B: E;aB
with
<
Bn '" ~ .
Conversely, suppose that (i)
=0
a.s.} , and verify that
U has the DC-property, let
To see that
S(t,r)
E
B
E
r}
Choose
t
E
U
S be fixed
£ > 0 with
S(t,£)
E
B,
is such that
P{E;aS(t,r) > 8} > 6
r
E
(0,£) , say for
r l , r 2 , ...
consistency theorem, there exist some random variables
such that
and we get by Fatou's lemma
By (i) and Kolmogorov's
E;O' ~l' E;2' .••
-62-
00
L ~n
contradicting the fact that, by (ii),
~ ~O <
00
a.s.
Hence (5)
1
can hold with fixed
so the set
{r
Eo:
This proves that
0 >
a
for at most finitely many
~S(t,r)
(O,E):
=a
a.s.}
r
E
(O,E)
must be dense in
,
and
(O,E)
U contains a base and hence has the DC-property.
Applying Theorem 5.3, it is seen that there exists some random measure
on
~
S
satisfying
kEN .
(6)
To extend (6) to
B, Bl , B2 , '"
E
B, let k
E
Z+
Bn
t
B.
B with
and
V
E
Uk
be fixed and let
By Theorem 4.4 in Billingsley (1968)
we get
~
d
V,B,B \ B
+
n
~
V,B,p~,
and so by continuity
d
~V B + ~V B .
, n
'
(7)
A similar argument shows that (7) is also true when
B
n
oj.
B .
On the
other hand
~(V,
Bn )
+
~(V,B)
so by the monotone class theorem in
B
E
B
a. s. ,
Lo~ve
B + B ,
n
(1963) we have
t;(V,B)
d
= ~V , B
Proceeding inductively, we obtain the desired extension of (6).
'
-63-
Turning to the existence problem for point processes, let us first
consider a necessary condition.
llA~(B)
Similarly, let
llA
= ~(A
A,B
~(B)
u B) -
~(B)
... llA
~
Define for any set function
E
B
on
B •
be the set function obtained by forming
n
1
B.
successive differences with respect to
~
Say that
is corrpletely
monotone if
n EN.
LEMMA 5.5.
Let
be a random measure on
~
fixed.
Then the set function
PROOF.
Let us first suppose that
Ee
-t~B
t
B
,
and let
S
~
E (0,00]
be
B , is completely monotone.
E
For the first order difference
= 00
we get
os
P{~A >
0,
~B
= O} = P{~B = O}
= P{~B =
O}
P{~A
= 0,
P{~(A u
B)
~B
= O} =
= O} = -l:IAP{~B = O}
,
and continuing inductively, we obtain
o
s P{ ~A 1 > 0, .,. , ~ n > 0, ~ B
In the case
t
<
00
,let
n
= O} =
n
(-1) l:I
... llA
n
Al
be a Cox process directed by
that
p{n B =
P{~B
o}=
Ee -t~B
B c
B .
= O}
t~
n
and note
EN.
-64-
Now suppose conversely that
U + au.
Unl ' ... ,Unk
be a null-array of nested disjoint partitions of
non-random points
s .
E
U .
j
,
= 1,
... ,k
.
n
nJ
nJ
S with all its mass confined to snl'
process on
U, U , U ' ... E U
2
l
1 whenever
V E U be arbitrary and let
Let
n
~(Un) +
U and such that
defined on the DC-ring
with
is a completely monotone set function
~
V.
For fixed
Let
~n
n
n EN, choose
be a simple point
, snk
and such that
n
(8)
= 1,
r{1; {s .}
n
nJ
j
E
J;
~ n {s nJ.} = 0,
= P{E; U .
n nJ
= {j~J
for each subset
J c {I, ... ,k } .
J} =
j ¢
0, j
>
J; 1;
U U .
n j¢J nJ
= O} =
[-OUnj]}~[j~J Unj ]
Note that
n
E
~
<"n
exists, since the
right-hand sides of (8) are non-negative, and summation over
J
yields
(The latter fact is easily established by induction in the number
1.
of partitioning sets.)
By (8),
U
where
with
Un
U
is the ring generated by
replaced by
U,
= U Un
Un
E
n EN,
Unl ' ... ,Unk
If
is known to be tight, it follows
{1;}
n
n
' so (1) is satisfied
n
as in the proof of Theorem 5.3 that (2) is satisfied for some point process
1;
on
S.
THEOREM 5.6.
Since (2) remains true for
Let
~
1; * ,
we reach the following result.
be a set function on some DC-ring
there exists some simple point process
E;
on
U c B.
Then
S satisfying (2) and such
-65that
Uc B
S
iff
(i)
~
(ii)
~(Un) ~
(iii)
for every
is completely monotone on
1 whenever
V
E
U
3
U, Ul , U2 ' ...
E
U with
Un
U there exists some null-array
~
au ,
{Unj }
C
U
of nested disjoint partitions of V such that the sequence
of point processes
sn
defined by (8) is tight.
NOTES. Theorem 5.4 is due with an entirely different proof to Harris
(1968) and (1971). For point processes, a much more general result (containing in particular the point process case of Theorem 5.3) is given by
Kerstan. Matthes and Mecke (1974), page 17. See also Prohorov (1961)
and Mecke (1972) for further general existence criteria. Finally, Theorem
5.6 is contained in the general results by Kurtz (1973) and Karbe (1973),
(cf. Kerstan, Matthes and Mecke (1974), page 35).
Our approach via Lemmas 5.1 and 5.2 seems to be new, although the
idea to prove existence by means of tightness arguments is well-established
in other fields, (see e.g. Billingsley (1968)).
PROBLEMS.
5.1.
Prove Lemma 5.2 and the corresponding result for diffuse random
measures.
5.2.
Show that, in the particular case of non-random
t;v' Theorem 5.3
reduces to a version of Caratheodory's extension theorem.
~66-
5.3.
Prove directly from (i) and (ii) of Theorem 5.4 that (ii) holds
with any finite number of terms. By (i) and Kolmogorovts consistency
theorem, PI;;-l can be consistently defined even for V E Ba>
Show
V
by means of (iii) that (ii) then extends to infinite sums.
5.4.
Suppose that
and let
~
S is discrete, i.e. that all its points are isolated,
be a set function on U = B
Show that (2) holds for
some point process
<PC"')
5.5.
=
1
on
I;;
S
iff
<jl
is completely monotone and
.
Show that Theorem 5.6 is false in general without condition (ii).
(Hint: let
Let
5.6.
S
=R
and place a fictitious atom at the point
U be the ring generated by all intervals
0+.
(a,b].)
Show that Theorem 5.6 is also false in general without condition
(iii) .
(Hint: let
I;;
be a stationary "point process" on the unit
circle, whose atom positions have exactly one limit point.)
5.7.
Show by a direct argument that the set function
is completely monotone for any random measure
6.
Ee
-tl;;B
,B
and fixed
I;;
E
B ,
t
<
a>
NULL-ARRAYS AND INFINITE DIVISIBILITY
In this section we consider the general central limit problem for
random measures, i.e. the problem of convergence in distribution of sums
L. I;; nJ.
, where the
J
I;;nJ· ,
in the sense that the
(1)
I;;nj
j,n
N • form a nuZZ-appay of pandom measupes
are independent for fixed
lim sup P{I;; .B >
j
nJ
n~
£
E} =
0
E >
0
B
n
E
B.
and such that
-67Note that, in the particular case of point processes, (1) reduces to
lim sup
n-+<>o
j
P{~
.B
>
nJ
O} = 0
\Lj ~ nj
tributions.
A random measure
for every fixed
E
B.
R, the class of limiting distributions
Just as for distributions on
of the sums
B
coincides with the class of infinitely divisible dis-
~
is said to be infinitely divisible 4 if,
n EN, we have
for some independent
and identically distributed random measures
process case, we require
~1'
...
'~n
In the point
~1'
to be point processes.
Note that,
since a point process may be regarded as an N-va1ued random measure, we
have two notions of infinite divisibility for point processes, (which
are not equivalent since the measure
in
M\N).
n~
may belong to
N even for
~
If nothing else is stated, we mean infinite divisibility as
a point process.
In our derivation of canonical representations and convergence criteria,
we shall proceed in steps, beginning with the simple particular case of
R -valued random variables.
In this case, (1) takes the form
+
lim sup
n-+<>o
We shall use
jections
K
j
.
nJ
> £}
=0
to denote the function
£
1 - e-
>
x
0 .
Furthermore, the pro-
are defined by
and
= x··
4
p{~
t
For a different notion of infinite divisibility, see O. Kallenberg,
Infinitely divisible processes with interchangeable increments and
random measures under convolution, Tech. Report, Dept. of Mathematics,
Gtlteborg, 1974.
-68-
LEMMA 6.1. An R+-valued random variable
- log Ee-~t
(2)
for some
and
a
A are unique.
may occur in (2).
then
L.~
{~
. &some ~
M(R')
E
+
L'P~
J
-1
w
. -+
nJ
, a and A satisfy
variables, (2) holds with
equivalent to
L. p~ -1.
J
nJ
t)
with
AK <
and
a
~
t
~ ,
0 ,
and in that case
A with the stated properties
is a null-array of R+-valued random variables,
.}
nJ
\'
K
~
A(l _ e
iff
J nJ
and then
+
Moreover, any
If
(3)
A
is infinitely divisible iff
-'IT
= ta
and some
a ~ 0
~
some
(2).
In the case of Z+-valued random
= 0 and bounded
a
w A on
A E M(N) , while (3) is
N
-+
Note that (3) is equivalent to the conditions
L'P~-~
~ A on R'+ with
J nJ
(i)
lim limsi
(ii)
e:+O
where
E
E:
+
=
,
(0 ,~J
E
~
.
e: nJ
limsi
=0 ,
=a ,
stands for both
limsup
and
denotes the truncated expectation at the level
PROOF.
(4 )
R'
L
j
n~
A{~}
Writing
~
~n
. = L~",.
nJ
,we claim that
nJ
= ? (1
- ~nj)
\'
~
e: > 0
d
f "'nj
liminf , and
-+
some
iff
J
-+
some continuous
~ ,
J
and that in this case
~ = L~ = e-~ .
In fact,
\'Lj ~ nj ~ ~
implies
-69-
II .4> • ....." , whence
J nJ
- Ij
~
and here
log ~ ..... - log
nJ
is continuous since
clearly equivalent to
max(l -
4>
j
sidering a Taylor expansion of
(4) implies
is an L-transform.
4>
a ,
.) ....
nJ
log
4>
~
n (t)
t
~
4> ,
~n
1\
-tl; .
Ee
nJ ) =
(1 - e-tX)~ (dx) =
n
~
J0 l-e -x
(K~n)
If
o l-e
-x
(t + 1) - ~ (t)
n
~(l)
= ta
(d) (dx)
~ ~(O)
R+
+
and note
? J (1
(dx)
_ e-tX)pl;-:(dx)
nJ
=
.
+
f
('1 - e -tX). ~ (dx)
t
~
o,
0
Conversely, suppose that
~,
and note that
-
e- (t + 1 ) X) ~n (dx)
= f (e - tx
o
=J
e
-tx
(KA) (dx)
o
, we may apply the continuity theorem for L-transforms
to the probability measures
weakly on
R
<Xl
f l-e -tx
(4) holds for some continuous
n
n E·N , on
given by
1/1
being continuous by monotone convergence.
~
J
<Xl
-tx
l-e
<Xl
+
nJ
J 0
<Xl
Hence (3) implies (4) with
= ta
j
-tl; .
L E (1 '" e n J ) =
<Xl
~(t)
= ~L p~-l.
"',
j
j
f0
LI;
. & some
. nJ
and it follows that
0 ,
= L (1 =
Conversely, by a similar argument,
.
IT 4> ..... some continuous
j
nJ
Since (1) is
(4) follows from this by con-
nJ
We next introduce the measures
that for
=~ ,
4>
K~ /~
n
n
K , and conclude that
But this is also true when
~(l)
= ~(O)
KA
n
converges
, since then
I;
-70-
The uniqueness of
a
and
in (2) follows from the formula
A
00
1/J(t + 1) - 1/J(t) =
f e -tx (a8 0
+
KA)(dx)
o
and the uniqueness theorem for L-transforms.
of
is possible, use (3) with
and
a
construct a
~
+
with
~
An
= 0 and arbitrary
a
divisible L-transform
=
¢n
n
= [n -1 ,
oo)A ,
for every
j = 1, ...
n
¢
E
E
N
~
by
~
in (2) follows
Conversely, any infinitely
can be difinition be written in the form
= L~
N
n EN, to
A , and then replace
In particular, the infinite divisibility of
a.
by dividing this relation by an arbitrary n
¢
To see that every choice
so the above argument applies with
~
,n , showing that
¢nj
= ¢n
has indeed the representation (2).
The
last assertion follows easily from (3).
We state explicitly the corresponding tightness criterion.
LEMMA 6.2.
Then
{I.~.}
J nJ
(i)
{~nj}
Let
is tight iff
limsup
n-+OO
(ii)
If the
~nj
L Et~
j
lim limsup
t
be a nuZZ-array of R+-vaZued random variabZes.
-+00
n-+OO
.
<
t
00
>
0 ,
=0
.
nJ
L. P{~ nJ.
t}
>
J
are Z+-vaZued, then (i) may be repZaced by the condition
(i)' limsup
11+00
I
j
P{~ . > O} <
nJ
00
•
-71-
PROOF. The sequence
R+
on
holds.
n
(KA n ) rO,t]
iff
of the last proof is vaguely relatively compact
{d }
t > 0 , i.e. iff (i)
is bounded for each fixed
The role of (ii) is to prevent mass from escaping to
thus
+~,
ensuring every vaguely convergent sub-sequence to be weakly convergent.
For point processes, (i) and (ii) imply (i)', which in turn implies (i),
so in this case, (i)' may replace (i) .
We now turn to the m-dimensiona1 version of Lemma 6.1,
~
Let
xt
=
x • t
LEMMA 6.3.
for the inner product of
E
R: '
x
m
a
E
m
R+
and in that case
and some
a
AE
A are
and
and
Rm
+
write
is infinitely divisible iff
m
-1
M(R+\{Ol) with (A~t)K < ~
unique.
Moreover, any
{~njl
If
a
and
is a null-
A as above such that
(i)
L·P~-~XA on ~\{O} ,
J nJ
(ii)
lim limsi L E ~ .{kl
£+0 Jl+<X> J. £ nJ
and then
E
Lj~nj ~ some ~ iff there exist
array of R:-valued random vectors, then
a
x, t
t
A with the stated properties may occur in (2).
some
and for
+ '
and
~
An R+-valued random vector
(2) holds for Bome
t
m
R
be a one-point compactification of
mEN
E;, ,
a
and
vectors, (2) holds with
= ark}
satisfy (2) •
A
a
k
...
,m
m
In the case of Z+ -valued random
= 0 and bounded
replace (i) and (ii) by the condition
= l,
A
E
M(Zm\{Ol) , and we may
+
L'P~-~ ~ A on Zm\{O} .
J
nJ
+
-72-
PROOF.
Let
a
A with the stated properties.
and
{~
.}
be a null-array satisfying (i) and (ii) for some
nJ
Then it is easily seen that
so by Lemma 6.1,
-TI
I.
E exp(-
t~ nJ.) ~ exp(-ta - A(l - e t))
J
and it follows hy the continuity theorem for m-dimensiona1 L-transforms
d
I·~
~ some
J nJ
.
that
~
satisfying (2).
d
I.~
~ some
J nJ
.
Conversely, suppose that
exist for every sequence
a
and
through
N'
c
~.
By Lemma 6.2, there
N some subsequence Nil
C
N'
and some
A the stated properties such that (i) and (ii) hold as
Nil.
that
n
~ ~
By the direct assertion of the present lemma, it follows
and
satisfy (2).
Now
at
is unique for every
by Lemma 6.1, and therefore a must be unique.
m
1 := (l, ... ,1) E R and note that by (2)
As for
A, write
+
Applying the uniqueness theorem for m-dimensional L-transforms, it follows
-'IT 1
that the bounded measure
(1 - e
-)A
is unique, and hence so is
A
-TI
since
1 - e
1:..
>
0
on- R:\{O} .
pendent of the choice of
n
+
00
through
This proves that
a
and
A are inde-
N' , and hence that (i) and (ii) remain true as
N, (cf. Theorem 2.3 in Billingsley (1968)).
assertions now follow as in the proof of Lemma 6.1.
The remaining
-73Before proceeding to arbitrary random measures, we prove a lemma.
LEMMA 6.4.
Let
semiring.
Then
kEN .
~
be a random measure on
is infinitely divisible iff
that
~V
,I k E I ,
II'
B be a DC-
is for evepy
be infinitely
kEN.
The necessity of our conditions is obvious.
~V
c
In the point process case it suffices that
divisible for every
PROOF.
I
and let
S
Conversely, suppose
for every V
is infinitely divisible in
E
I
k
,
kEN .
Since infinite divisibility of random vectors is preserved under convergence
in distribution, (as may be seen from the continuity theorem for L-transforms),
we may use the monotone class theorem in
to arbitrary
V
E
k
B
,
kEN
~V
n E N some random vectors
(L~V)
(5)
-n
= L~
Lo~ve
(1963) to extend this property
It follows that there exist for every
,n
satisfying
k,n EN.
V,n
From (5) it is easily seen that the family
{~V
, n}
satisfies the conditions
in Theorem 5.4, and hence there exists some random measure
~ nV ~ ~V , n '
k
VE B ,
k
E
L~V :: (L~ V)
N
n
~n
on
S with
Writing (5) in the form
k
k,n EN,
V E B
n
and using Theorem 3.1, it is seen that
~
is infinitely divisible.
Turning to the point process case, suppose that
divisible (as a random variable in
kEN.
Z+) for every
I·o.
J J
is infinitely
II'
Since repetitions of I-sets may occur, this proves that, for
-74-
any fixed
V = (11'
divisible in
Z+
...
,I k )
for every
I
(
k
, kEN , tW = L·t.f;!.
...
t = (t l ,
,t k )
J J
Nk
€
R+
for every
=
t
is infinitely
Turning to rationals
and using the above closure property, it follows that
infinitely divisible in
J
tt;V
(t l , ... ,t k )
is indeed
€
R~.
Hence,
A ;:: 0
by Lemma 6.1, there exist some constants
p
such that
= L (1
- log Ee-utt;V
p
_
e-utp)A
p
In fact, there can be no constant component
infinitely divisible in
and hence
Z+
here since the
a
P{t;I.
J
= O}
>
0 .
Theorem 6.9 below.)
vector in
Z:
pt;-l.
Ltt;V
above, and so
it follows that
Since
tn ~ tt;V
.
A is
n is an infinitely divisible random
A=
with spectral measure
LpA p
are
(This is a simple special case of
Now suppose that
is ensured by Lemma 6.3 since
J
Moreover, making
a Taylor expansion in (2), it is easily seen that the support of
contained in the support of
t;I.
<
\L.p AP0p
, the existence of which
Then
00
t.
If the
J
n d:: t;V , which proves that
coincides with
L
tn
are chosen rationally independent,
t;v
is infinitely divisible.
V was arbitrary, we may now proceed as in the first part of the
proof to conclude that
~
itself is infinitely divisible.
We are now ready to extend Lemmas 6.1 and 6.3 to random measures
defined on general spaces
THEOREM 6.5.
(6)
fi
S.
~
pandom measure
Log
Ee -H
'
= af
on
S is infinitely divisible iff
-1r
+
A(l _ c
f)
f
E
F ,
-75-'
for some
B
E
a
E
M and some measure _ A on
B , and in this case
and
a
a
and
Moreover~
A are unique.
with the stated properties may occur in (6).
S ~then
ofY'andom measures on
satisfying
M\ {o}
If
I.F;.
. ~ some
J nJ
any
a
and A
bS a null-array
{E;,nj}
iff there exist some
t;,
A as above such that
\
KL'P(t;, .f)
J
nJ
(7)
and
then
t;,
~
a
aaI
= A{~aI
>
O}
I . P(t;,
(8)
J
and
-1 w
~
afo O +
-1
K(A~f
=0
1 E I ,
.Y) -1 v
nJ
-1
A~V
~
I
c
E
F
c
B is a DC-semiring satiSfying
then (7) is equivalent to the conditions
on ~\{o}
+
lim limsi L E t;, .1
n-+<>o j E: nJ
(9)
~f
A satisfy (6).
f
)
y E
al
=
rk
kEN ,
1 E I
E:~
In the point process
to
N\{O}
case~
and satiSfies
(9) can be replaced by (8)
we have
a
= 0 in
-1
(A~B )N <
alone~
00
E,Y
Suppose that
B.
B E
is confined
A
In this
case~
(8)
and
z:\{O}
t;,
is an infinitely divisible random measure. Then
k
is infini tely divisible for every Y = (B , ... , Bk ) E B , kEN
l
so by Lemma 6.4 there exist some unique
~
E
such that
(10)
while
provided the vague convergence on It\{O}
+
is strengthened to weak convergence on
PROOF.
(6)
- 1og Ee -t(t;,V)
= t ay
+ A~ y (1
-~
- e
t)
R:
and
Ay
E
M(R:\{O})
-76-
Choosing all components
t{j}
but one equal to
0, we obtain
a y = (a
,a ) , while choosing exactly one component equal to
'
Bk
B1
a , it is seen that {Ay } is in the obvious sense a consistent family
of measures.
and
Let us now choose
Y
=
(A,B, A u B)
B are arbitrary disjoint sets in
= AA , B{(x,y):
A'C
we get for any
t
=
= _
A--Cl + .-V
Defining
(x,y, x + y)
by
C} ,
E
E R+
e
t) -_ _ log Ee-t(~V) -_
log Ee-Cu+w)~A-(v+w)~B
(u + w)a
=
A
3
(u , v , w)
-7T
ua A + va B + wa AuB
B.
in (10), where
A
+ (v + w)a
= uaA + va B + wCa A +
B
+
J (1
- e-Cu+w)x-(V+W)YPA,BCdXdy)
-7T
a ) + A'(l - e
B
t)
By the uniqueness assertion in Lemma 6.4, it follows that
(11)
and
A
V
= A' '
and in particular
(12)
Next suppose that
that
-t
a
B , B , ...
l
2
while
= A'{(X,y,z)
E
3
R+ : x + y ~ z}
= AA , B{(x,y)
E
R::
E
B with Bn (~
~
=
x + y ~ x + y}
=0 .
, and conclude from (10)
-77(13)
> 0 .
E:
Combining the former relation with (11) , it is seen that the set function
B -+
B
f
, B E B is in fact a measure. As for the family fA }
v
B
k
k
B be fixed and define the measures A'V E M(R ) , V E B , k
+
let
Cl
Since
{A V}
E
N , hy
carries over to
,
we may apply Theorem 5.4 to the latter family, and conclude from
(12) and (13) that there exists some unique measure
satisfying
Defining the
AB(d)
IJ
mea~ure
A'
B
on
{IJB >
O}
E
on the same set by
- e- IJB ) -lA' (d1l)
= (1
B ~
we ohtain for any measurable set
f
(1
-- A
B,B,V
-x -1
)
(R' 2x C)
+
and it follows that for any
This proves that the measures
(1 - e
-y
= AB,V (R'+
B'
E
)A
x
-x -1
)
JR'+
B,
B V(dxdy x C)
{IJVEC}
(1 - e
(1 - e
=
C)
,
An v(dX
,
x
C)
=
=
'
B with B'
~
B
AB are all restrictions of
A = sup 1\B .
BEB
M
-78-
It is now easy to verify that this
(6).
A is the unique measure satisfying
Using Theorem 5.4, it is further seen that any
a
and
A with
the stated properties define the distribution of some random measure
~
satisfying (6), and it follows in particular that any such
is infinitely
~
divisible.
{~nj}
Now suppose that
(7) for some
is a null-array of random measures satisfying
A with the stated·properties.
and
measure satisfying (6), it follows by Lemma 6.1 that
d
2.~
~~.
J nJ
.
and hence by Theorem 4.5,
L.~
. ~ some ~. Then
J nJ
~
is a random
L'~
.f ~ ~f,
J nJ
f
E
Fc
Conversely, suppose that
is infinitely divisible by Lemma 6.4, and
hence it satisfies (6) for some
Lemma 6.1.
If
and
a
A.
But then (7) follows by
The same argument may be used to show that (8) and (9) are
necessary and sufficient, and therefore are equivalent to (7).
Finally,
the assertions for point processes may easily be deduced from (8).
We shall now give two useful and illuminating interpretations of
the canonical representation (6).
divisible random measure
write
satisfying (6).
lCa,A)
for any infinitely
In the point process case we
1(0,1..) = 1(1..) .
LEMMA 6.6.
Let
a
Poi8son process on
(14)
~
Let us write
and
A
M\{O}
be such as in Theopem 6.5 and let
with intensity
l(a,A)
~
a
+
Then
A.
f lJn(dlJ)
.
n
be a
-79~
PROOF.
Denote the 1eft- and right-hand sides of (14) by
Since for fixed
f
F the mapping
€
1T
f
~f =
and
~
: ].l
-+
].lf
is measurable, it follows
O'.f
+
n1T f
is a possibly infinite-
~
as in Section 1 that the function
~
valued random variable, and we obtain by (1.2) and (6)
(15)
Ee
-~f
= e -afEe -mTf = e -af
d
~
In particular
= ~B
~B
<
a. s. ,
00
(
-1r f )
-~f
exp -A(l - e
) = Ee
.
B E
B , and it follows as in Lemma
~
1.3 that
is a random measure.
~
Finally, (14) follows from (15) by
Theorem 3.1.
LEMMA 6.7.
Let
and
0'.
A be such as in Theorem 6.5, and let
be a disjoint partition of M\{O}
n ( N , we further assume that
mean
AM
n
such that
vn
and that
common distribut'ion
AM
n
<
00
,
n EN.
n
n
•
For each
is a Poissonian random variable with
are random measures on
M VAM
M , M2 , ...
l
Mh
with the
Letting all these random elements be in-
dependent, we have
I(a,A) d a
(16)
PROOF.
+
The right-hand side of (16) has the L-transform, evaluated at
an arbitrary
f
E
F ,
-80-
e
_af
-AM
00
II e
n=l
=
e
= e
-af
-af
00
00
e-a.f
~
n
k=O
II
n=l
00
II exp{ - (M A) (1
n
n=l
-'IT
exp{-a.f - A(l _ e
exp{ -A (1
f)}
and by Theorem 6.5, this agrees with the L-transform of
1 (a,A) .
Thus
the assertion follows by Lemma 1.3 and Theorem 3.1.
The last two lemmas may be used to establish various relationships
between an infinitely divisible random measure
canonical measures
THEOREM 6.8. Let ~
a.s. iff (i)
s
E
S .
a
;t
~ E
* = 0 a.s.
E
P{~:
Sand
;t
O}
a
~ p{~{S} ~
*
~a = 0
o} = o .,
E
Md and A~ = 0 •
By Lemma 6.7 we obtain (i), and
AMn
n EN, and summing over
E > O.
Then
O} = 0 ., and (iii) "A{ll{S} >
*
snJ a = 0 a.s. whenever
(iii), suppose conversely that
s
;t
Md a.s. iff
~a
a > 0
S and let
on
( ~.)
O} = 0,
and the corresponding
We shall give two results of this type.
o ., (ii) A{l-! *a
Suppose that
furthermore,
A.
g 1 (et,A)
In pa:I'ticular.,
PROOF.
M A{l-!
n
a
a* =
and
a
~
~A{l-!{S}
>
>
n
O.
But then
yields (ii).
E} > 0
for some
To prove
n EN,
Then we get the contradiction
a}
~
p{Vn >
• P!Vn
[a/E] ,
>
[alE)}!
~nj{s}
>
P{Enj!S}
E , j
>
S[alE]
Ellla/EJ.1
+l} =
>
0 •
-81-
so (iii) must indeed be true.
Conversely, suppose that (i) - (iii) hold.
that
a.s. whenever
AMn
>
From (ii) it follows
0 , so it suffices to prove that,
wi th probability one, no two terms in (16) can have atom positions in
common.
But this follows by Fubini's theorem from the fact, implied by
(iii), that
~
.C = 0
nJ
a.s. for any countable set
SeA)
To state the next result, let
A on
c
S .
denote the support of a measure
M w.r.t. the vague topology in M. Moreover, given any M eM,
define
write
C
GM
M generated by M u
as the additive semigroup in
a = min M whenever
THEOREM 6.9.
Let
t;
a
M and
E
d
= l(a,A)
on
a
~ ~,
and
{a} ,
~ EM.
Then
S.
(17)
(18)
GS(A)
PROOF.
= S (P t; -1 )
-
Ct
•
To avoid trivialities, we may assume that
consider the case of bounded
(19)
~
A
d
v
L
a +
j=l
is Poissonian with mean
AM
t;. ,
J
while the
v
of
and mutually independent with distribution
is seen that
~ ~
a
a.s. and
addition and subtraction of
P{t;
Ct
Let us first
By Lemma 6.7, we may then assume that
where
v
A ~ O.
=
a} = e
-AM
>
t;.
J
are independent
A/AM.
From (19) it
0 , proving (17).
are continuous operations on
M and
Since
~82-
M+ a
respectively, we further obtain
- a
and so it suffices to prove (18) in the case
~O
Suppose that
~l'
and
'~k
...
d
P{p(E,;,\1 ) <
0
E >
0
P{v = k
~
=
~l + ••. + ~k
By the continuity of additien in
E SeA)
exists for every
~O
E GS(A)\{O} , say
a = 0
some
0
<5 >
, P (~j ,
~.
J
,where
kEN
M, there
such that
, j = 1,
) < 0
...
,k} =
k
= p{v = k}
P{p (~ . ,
II
j=l
e
=
GS(A)
c
) < o} =
A{p ell, \1')
II
J
j=l
~O E S(p~-l)
proving that
J
k
-AM (AM)k
k!
~.
J
< o} /
Sep~-l)
Since
AM
>
0 ,
is closed, this proves that
S(p~-l) .
110 E S(p~~l) .
Conversely, suppose that
obtained as
~
v
in (19) but with
For
replaced by
n EN, let
v An.
Then
a.s., so we get
(20)
o<
P{p(~,~O) < £} ~
liminf P{p(nn' llo) < £}
£
>
0 ,
n~
and hence there exist some
we can show that
k
E
S(p~-l)
-1
m E S (Pn n )
{O,l, ... ,n}
, n
lln E GS (A)
proving the relation
assume that
n EN, with
c
€
N
GS(A)
it will follow that
Keeping
If
\1
n E N fixed, we thus
Then there exists for every pEN
satisfying
0 E GS(A) ,
some
-83-
o<
(21)
P{p(~, m) < p
-1
, \I
.
= k}
t::
P{P(~l + ... + ~k' ID) <
=
P{P(~l + ••• + ~k' m) < p
and since the set of possible k-va1ues is non-increasing in
choose
k
independent of
p.
Moreover,
so we can further assume that
~
o.
implies
1
-1
-1
= k}
, \I
= k} ,
}P{\I
p, we can
m
=0
E GS(A) ,
Let us now define the sets
k
-1
, llk) EM: p (11 1 + ••• + llk' m) < p }
Ap = {( 11 , ...
Since
k
=0
k
P
is separable in the product topology, each
pEN .
A
p
can be covered
by countab1y many sets
j
with
(22)
(m 1 , ... ,mk ) E A ,
p
o<
80
satisfying
P{p(~,
< p
JP
k
-1
~oo
=
j
we get
~
JP
I, ... ,k}
lim (m 1
P
~
+ •••
showing that the sequences
m
that
m1p
m.
JP
+
...
Let us now fix
v m.
~
J
+
+ mkp)f = mf <
j E
II
P{p(~.,
J
U,
...
+
.. .
,k}
N'
c
f E F
< P
JP
p
~
00
,
-1
,k
so
and
E:
> 0
We
m , ... ,mk E M
I
It follows in particular
(p E N' ) , so
mk
}.
j = 1, ... ,k ,
c
N and some
= 1,
+
00
v
m ~
m as
kp
m. )
,{mkp } are relatively compact.
{m lp }'
(p EN') , j
m v~ m
kp
1
+ ... +
1p
may therefore choose some sequence
such that
=
j=l
From
1imsup m. f
I, ... ,k}
by (21) there must exist some vector
,mk ) E A
p
p
m. )
=
m
, and choose
= ID1
+
...
pEN'
mk
so large
+
-84-
that
p (m.
JP
, JD.) <
J
P{p(l;., m.)
J
J
<
d
£/2
and
P
~ P{p(~.,
-1
m.)
J
£/2 .
<
JP
By (22), we then obtain
< E:/2} ~ P{p(~.,
m.)
This shows that
m
J
so we get
completes the proof in the case of bounded
In the case of unbounded
A
p -1 } > 0
<
JP
GS(A)
E:
A .
we still have
6.7, and if (18) is true, we get in addition
a
~ ~
S(P~
E:
a.s. by Lemma
a
-1
) , proving (17).
It thus remains to prove (18), and again we may assume that
and
~nj
v
Then
~n +
~
v
=
= L L
~n
~kj
k=l j =1
~O E
some
~n E:
that
~.
As
J
M1
Z
E:
+
~
E:
N .
N,
E:
A are bounded and form a non-decreasing sequence. Suppose
n
S(p~-l)
Proceeding as in (20) , it is seen that there exist
v
S (p~-l)
n E: N , with ~n + ~ , and since ~l' ~2' ... E: GS (A)
n
~O
Conversely suppose that
k
For
a.s., and moreover,
by (23), it follows that
some
n
,
n
that
= O.
k
(23)
since the
a
as in Lemma 6.7, define
n
A
n
and hence
a
and
~l'
for each
...
j
GS(A) .
E:
~O E:
'~k E:
E:
U,
GS (A)
~
,k} , say that
in Lemma 6.7 we can choose
k
Ml = U
j =1
{~ E
M:
a = ~l
...
+
+ ~k
for
Here we may assume without loss
SeA)
. ..
, say
~f.
J
1
>2~·f.}
J J
~.f.
J J
;t
a
where
f.
J
E:
F
c
-85-
since by Theorem 6.5
k
~
AMI
The set
L
j=l
Ml
A{~f . >
J
1
'2
k
~.f.}
J J
-1 1
00) < 00
AlT f. (2~·f.,
J J
I
c:
j=l
J
~l'
being open and containing
,]1k ' we get
and hence by (23)
(24 )
Let
£
>
0 be arbitrary, and choose
(25)
p(]1, ]10) v p(]1I, 0) < 0
0
=?
which is possible since the mapping
Since
~ - ~n
X0
= P{p(~
£
€
(0, £/2)
p(]1
+
(]1, ]1')
~
-
n
, 0)
By (24) - (26) and the independence of
and since
€
-
~n'
0)
< c} >
t"
':on
p(j1
+
This completes the proof.
n
+
]1',]1)
is continuous.
N such that
€
0 .
and
~
< o}P{p(~n'
was arbitrary, it follows that
such that
]1', ]1) < £/2 ,
a.s., we may further choose
P{p(~
(26)
N .
n
]10
€
-
~n
]10)
g(P~
' we obtain
<
-1
&} > 0 ,
) , as desired.
-86-
NOTES.
The canonical representations occurring in Lemmas 6.1 and 6.3
are classical, (see e.g. Feller (1968) and (1971)), while the one in Theorem
6.5 is due to Matthes (1963) and Lee (1967), (see also Mecke (1972)).
The
corresponding convergence criteria are due to Kallenberg (1973a), (though
the proof of Lemma 6.1 follows that of Feller (1971) for general random
variables).
As for Lemma 6.4, the first part is given in Kerstan, Matthes,
and Mecke (1974), page 49, while the second part is new.
Lee (1967) gave a
proof of Lemma 6.7 along with a heuristic argument for Lemma 6.6; the
presentproof was supplied for point processes by Jagers (1973).
Finally,
Theorem 6.8 extends a result for point processes by Kerstan, Matthes and
Mecke (1974), page 95, (see also Kallenberg (1973a)), while Theorem 6.9
is new.
The reader should consult Kerstan, Matthes and Mecke (1974) for
a different approach to the theory of infinitely divisible point processes,
where (16) plays the role of a basis for the whole theory while (14) and (16) fit
into their general theory of "showerfields".
PROBLEMS.
6.1.
Prove the Z+-version of Lemma 6.1 directly by means of generating
functions.
6.2.
State and prove tightness criteria corresponding to the convergence
assertions in Lemma 6.3 and Theorem 6.5.
6.3.
Let
6.5.
F;,
d
= l(a
n E N , in Lemmas 6.1 and 6.3 and in Theorem
n
n' An )
Give criteria for convergence in distribution of {F;,n} in
terms of
6.4.
{a }
n
and
{A }
n
Assume that S is countable. Show that there exists some fixed
f E F such that, for any point process F;, on S with F;,S <
00
a.s.,
F;,
is infinitely divisible iff
F;,f
is infinitely divisible
-87 -
vanishing constant component.
(Hint: use the result in Problem
3.8.)
6.5.
Let
E,
be a point process on
divisible iff
for every
6.6.
For
E,
f
E,f
E
I(O,A)
=
S.
Show that
is infinitely divisible with
pE,-l
is infinitely
P{E,f
= O}
>
0
F
c
on
S with
AM
<
00
state and prove a relationship
,
between the set of atom positions of
also that
E,
pE,-l
can have no atoms if
be strictly positive and such that
=
AM
E,f
<
00
and that of
A.
Show
(Hint:
Let
f
00
a.s.
E
F
Then
-1
R~ = A{~f > O} = AM , so the last assertion follows from the
f
corresponding fact for random variables due to Blum and Rosenblatt,
ATI
see Lukacs (1970), page 124.)
7.
FURTHER RESULTS FOR NULL-ARRAYS.
In this section we shall see how the results of Section 6 can be
improved in certain special cases.
Let us first consider the case of
random measures with independent increments, (sometimes called completely
random measures).
[,B , ... ,E,B
k
l
kEN .
LEMMA 7.1.
b;. 0
Let
E,
are independent for arbitrary disjoint
Let us say that a measure
for some
iff
By this we mean random measures
and
~ E
such that
B , ... ,B
k
l
M is degenerate if
~
E
B ,
= ho s
s c S .
E, ~ l(a,A)
on
S.
Then
E,
has independent increments
A is confined to the class of degenerate measures.
-88-
PROOF.
Suppose that
kEN
A has the asserted property, and consider arbitrary
and disjoint
for any
B ,
l
,B
k
E B.
Writing
= M\{O}
M'
, we obtain
t l , ... ,t k E R+
L t.~B.)
- log E exp(-
j
J
J
=
J
L t.aB.
J J
L
+
j
j
f
lJB.
J
=
I
{tjaB j
j
=proving that
;B l ,
... ,;Bk
L
t.llB.))A(dll)
j
J J
(1 - exp(-
t.nB.
J
+
(1 -
>0
J
log IT E exp(j
t.~B.)
J
J
are independent.
;
has independent increments, and let
a
is additive, we obtain as above
exp (- t j lJ Bj )) A(d II ) =
(1 - exp(- t. lJ B.))A(d J1 )}
J
M'
J
Conversely, suppose that
B E B be disjoint.
A and
f(x) :: 1 - e -x
+
y)
<
f(x)
+
fey)
is strictly concave on
for every positive
is only possible provided
)l
A{J1A
>
0 , )lB
>
S into countably many sets of diameter
E
M'
Since
a.c.
Since
o.
But the function
of
=
,
(1)
f(x
=
a.e.
€
A.
x
O}
and
= O.
< €
,
R
+
, so
y, and hence (1)
Making a partition
it follows that, for
A, at most one of these sets can have )l-measure
can be chosen arbitrarily small, this proves that
lJ
> 0
is degenerate
-89-
THEOREM 7.2. A random measure
~
S is infiniteZy divisibZe with
on
independent increments iff
= af
log Ee -H
(2)
f
+
f
F ,
E:
R txs
a
for some
+
y
M and some
E:
and in this case
a and
E:
with
MeR t x S)
+
yare unique.
Moreover, any
the stated properties may occur in (2).
random measures on
and if
S
I
c
B~
+
Ky (.
y(o x B)K <
{~
If
.}
nJ
~
B
,
B ,
E:
a and y with
is a nuZZ-arPay of
I·~
is a DC-semiring, then
. +d
~
] nJ
iff
\'
. P (~
(i)
KL
(ii)
I·p{~ .1
]
. I)
-1 w
+
nJ
a 10
A ~ .1
nJ 1
nJ 2
]
11' 1 2
E:
0
Tn this case it suffices to take
PROOF.
Let
I .P(~
]
~
£
£
0
=
on
whiZe
y
1.-S
confined to
->
(~S,
t
~
be the measure on
)
N
,
~.
I
E:
I
x
S
~
EM, let
From Lemma 2.1 it is seen that
is a measurable mapping of
R'+
S
N x
in (ii) and to repZace (i) by
M E M be the class of degenerate measures, and for
~
and disjoint
0
>
is infinitely divisible with independent increments.
the unique atom position of
that
a = 0
w
.1) -1 + y (. x I)
nJ
Suppose that
I , and
1 .
In the point process case we have
(i) ,
E:
for every
0
> £} +
I
I) ,
X
induced by
M into
Rt
+
A via this mapping.
S .
X
M E:
t
be
M and
Let
Then (2) is
obtained by transformation of the integral in (6.6), and since
~
y
-90 ..
y(dx x B) =
(3)
we have
A{~B €
= (A~B-1 )K
y(o x B)K
<
note that the inverse mapping
and (2) into (6.6).
a
and
y
dx}
x > 0
B
m
(xJt)
B ,
€
B . To see that
€
xO
+
B
transforms
t
is unique,
y
y
into some
A
From the above mappings it is further seen that any
with'the stated properties are possible in (2).
As for the convergence assertion, the necessity of (i) and (ii)
follows easily from (3) and Theorem 6.5.
and (ii) hold.
Conversely, suppose that (i)
Then (6.9) follows from (i), and it reamins to prove (6.8).
It is then enough to take the components
...
II'
,I
of
k
V
in (6.8)
disjoint and to verify that
k
k
L
(4)
j
for any
p
n
i=l
H: n].r.1
~
t.}
1
1
~
t.}
1
t.1
= 0 or
A{~I. = t.} = 0
1 1
with
Since events corresponding to
from (4), we can assume that all the
from (i) for
{~I.
n
i=l
t = (t ,
l
i = 1, ... ,k .
A
+
are
t.
1
t.1
=
> 0
k = 1 and from (ii) and Lemma 7. 1 for
0
can be omitted
But then (4) follows
k·~
2.
The last ,
assertion follows from the corresponding fact in Theorem 6.5.
The most general random measure with independent increments is obtained
from the one in Theorem 7.2 by addition of at most countably many fixed atoms:
COROLLARY 7.3.
A random measure
n on
S
iff it can be written in the foY'/11
k
(5)
n
= t;. +
L
j=l
S.Ot
] j
has independent increments
-91-
wheT'e
S
~
satisfies (2) faT' some
S , some
E
k
E
T'andom va:fliables
a
Z
and distinat
S.
with
+
J
P{S.
t.
J
:s;
> a} > 0
J
independent and independent of
and
y
with
j
S ..
,
j
a{s} = y(R'
+
k .. and some R+ -valued
:s;
:s;
{s}) = 0 ,
x
k
J
whiah aT'e mutually
In this aase .. the above deaomposition
~
is unique apaT't fT'om the oT'deT' of terms.
PROOF.
that
The sufficiency and uniqueness assertions are obvious.
has independent increments.
11
Like any random measure"
at most countably many fixed atoms, i.e.
many
s
and
~
E
I1{S}
be defined by (5).
= a a.s. for all but countably
Since
11
~
atoms, so if
{B .}
has independent increments.
nJ
c
Moreover,
null-array of random variables.
Bl , ... ,Bk
,~Bk)
E
~
~,and
has no fixed
{~B .}
nJ
is a
A similar argument shows more generally
is infinitely divisible for any
kEN
B , and so we may conclude from Lemma 6.4 that
infinitely divisible.
and disjoint
~
is itself
In view of Theorem 7.2, this completes the proof.
By comparison of (2) and (1.2), it is seen that a point process
~
on
S
J
Hence, by Lemma 6.1, the common row-sum
must be infinitely divisible.
(E:B l ,
t.
B is a null-array of partitions of some fixed set
B , it follows by a simple compactness argument that
that
Let
has independent increments, it is easily
are mutually independent and independent of
Sj
further that
~B
can have
k , be the corresponding atom positions and sizes, and let
:~
j
S· ,
J
E
11
S , (cf. the argument in Billingsley (1968), page 124).
seen that the
B
Next suppose
is Poissonian iff it is infinitely divisible with independent
increments and with the measure
y
in (2) confined to the set
{I} x S
-92-
In this case,
~
has intensity
E~B
= yeO}
B
x B)
B •
E
Specializing Theorem 7.2 to this case, we obtain the important
COROLLARY 7.4.
Let
and let
be a nuU-arTay of point prooesses on
that
I
{~nj }
c
B
~
be a Poisson process on
is a DC-semiring .
A
(i)
2'P{~ .1 > O} + AI
J
nJ
(ii)
I.p{~ .B > l} + 0
nJ
J
S with intensity
S
A EM,
Further suppose
I.~
. ~ ~ iff
J nJ
Then
IEI,and
B
E
B •
We are now going to consider the convergence of null-arrays towards
two special types of infinitely divisible random measures which play the
same role here as do the simple point processes and diffuse random measures
in Sections 3 - 5.
But first we define regularity for canonical measures
A and for null-arrays
C
c
B and any
a
>
{~nj}
0 , we shall say that
if there exists for every
of
as follows.
C
E
Given any covering class
A is a-regular w.r.t.
C some array
{C mk }
c
C
C of finite coverings
C such that
(6)
lim
]}t+OO
2 A{lJCmk
k
The notion of a-regularity of
(6 ) replaced by
{~nj }
?:
a}
=0
.
is defined analogeously but with
-93lim limsup
~
Ul'-
n-+oo
L. kL P{~ nJ.Cmk
~ a}
=0
"
J
As before, regularity means a-regularity for every
Let
THEOREM 7.5.
let
[, ~ 1 (A)
{~
I
(i)
lim L P{E, .U > O}
.
nJ
n-+<oo J
(ii)
limsup
(iii)
j
lim limsup
k-+oo
Moreover,
I
L P{E,
n-+oo
Lj~nj
n-+oo
d
~
+
=0
= A{~U
Then
>
\Lj ~ nj ~ E,
1
L P{~
.B
>
k}
nJ
=0
B
E
L EE,
j
if.'~J
<
I ,
B .
follows from (i) and (ii) if
n-+oo
is
B~
UE U ,
O}
.1 > I} ~ A{~1 > I}
j
limsup
uc
Further suppose that
and
S
nJ
and from (i) alone if {E,nj}
(7)
.
a DC-semiring.
B~
c
be a null-array of point processes on
nJ
A{~ ~ ~ * }
with
a DC-ring and
.}
a > 0 .
A
is 2-regular w.r.t.
.1 ~ A~I
<
00
I
E
&S
I
2-regular w.r.t.
or if
I .
nJ
The proof of this theorem is similar to that of Theorem 4.8, except
that we now rely on the following two lemmas, playing the roles of Theorem
3.3 and Lemma 4.7 respectively.
LEMMA 7.6.
Let
A
be the canonical measure of an infinitely divisible
point process, and suppose that
cerm:-ring.
7'hen
and in that case
A{~
A
~ ~*} =
&S
0
U c B is a DC-ring and I
iff
A{~1 > l} :0;
uniquely determined by
B a DC-
c
A{ll * 1 > l } ,
A{~U >
O}
U
1
E
E
U •
I ,
-94-
PROOF.
by
e
E
U , let
A via the mapping
~ + e~
For fixed
A
,
N' = N\{O}
be the measure on
e
induced
i.e.
MEN n N' .
Then
Ae
so either this quantity is zero, or
can be normalized to a probability
N' , i.e. to the distribution of some simple point process.
measure on
I.e is completely determined by the quantities
Hence, by Theorem 3.3,
U€ U,
UE U
Le. by all
for any
f
But
F with support in C
E
A
C
determines the measure
so by Theorem 6.5, the family
{A }
C
determines
A.
We may now complete the proof as in case of Theorem 3.3.
LEMMA 7.7.
Let
{E,;nj}
d
I.E,; . + som~
J nJ
suppose that
that
(8)
U
n·
Further suppose that
S , and
d
= I (a,A) and
E,;
B '[.,s a DC-ring satisfying
liminf
I
n-+oo
j
for some
~C = 0
c
be a nuZZ-array of random measures on
t
a.s.
-tE,;
(1 - Ee
Then
> 0
.U
n J ) ~ taU
nC = 0
-tn
+
1.(1 - e
u)
U EO U ,
a.s. for every compact
C
E
B with
In the point process case, (8) can be repZaced by
1iminf
n-+oo
I
j
P{~ .U > O} ~ A{~U > O}
nJ
U
t:
U.
-95-
This may be proved in the same way as Lemma 4.7.
We now turn to the analogue of Theorem 4.9.
THEOREM 7.8.
on
Let
respectively).
o<
~ ~
and let
S
be a nuU-arTau of point processes (random measures)
nnj}
Further suppose that
and that
U c BI;
Then
\L.j I;nj ~ I;
(i)
lim L
n-+oo j
(1 - Ee
(ii)
limsup
n-+oo j
s < t
semi-ring) .
(or' with
AM
I
c
BI;
.U
nJ )
-sl; .1
nJ )
>
t}
A(1 _ e
+
~ sal
U)
U
U ,
E
-STI
+
A(l _ e
I
= 0
E
I)
I
€
I ,
I .
~s 2-regular (regular)
$
aC
+
ATI
I
<
00
I E I
.
d
!;
Suppose that
and let
by Lemma 4.3, so by Lemma 6.1
U or
I
Then
0
-tw
taU
=
point process case is similar but simpler.
E
=
if.f
We confine our attention to the random measure case, since the
B
d
a covering class (DC-
follows from (i) alone if {!;nJ'}
limsup I EI; .1
n-+oo j
nJ
c
are fixed real numbers with
t
or if
I
(9)
PROOF.
(1 - Ee
sand
A{}l;t /} = 0
is a DC-ring and
lim limsup I P{~ .1
t+oo
n+oo j
nJ
r·t;J nJ. ~!;
Moreover.,
w.r.t.
-tl;
r
(iii)
a = 0 and
with
I (a,A)
L.I; .B &!;B
J nJ
\.1; .
L. J nJ
+
This implies (i) and (ii), while (iii) follows by Lemmas 4.6 and 6.2.
-96Conversely, suppose that (i) - (iii) hold.
is a decreasing function of
x > 0
r
and
> 0
(1 - e
-tx
)/x
we have
o :-:;
and so we get for any
Since
U
E
x
:0;
r ,
U
By Lemmas 4.6 and 6.2, it therefore follows from (i) and (iii) that
is tight.
But in that case every sequence
such that
Los,
J nJ
6.5, and since
d
+
some
Uu I
c
n
(n
B
E
Nil)
N'
Now
c
n
{LjSnj}
N has a subsequence
d
= some
Nil
by Theorem
by Lemma 7.7, it follows from (i), (ii) and
n
the necessity part of the present theorem that
-t7T
taU
+
~ (1
sal
+
~(l - e
e
U)
l)
-t7T
e
=
telU
+
1.(1
:-:;
sell
+
1.(1 - e
-S7T
U)
U
E
U ,
l)
I
E:
I
-S7T
,
which by Theorem 6.5 is equivalent to
(10)
Ee -tnU
= Ee -tsU
(11)
Ee -snl
~
Ee -ssl
U
E
U ,
I
E:
I
Furthermore, it follows from Theorem 6.8 that
s
is a.s. diffuse, possihly
apart from atoms of fixed size and location which arize from the atoms of
(1.
or
Let us now consider any fixed
1 respectively such that
B
n
XES
~
Choosing
Bl , B , ...
2
in
U
{x} , it follows by monotone convergence
-97-
from (10) and (11) that
(12)
Ee -tnJx}
(13)
Ee-sn{x} :<: Ee-s~{x}
If
n{x}
= Ee -tHx} = e
=e
-ta{x}
-sa{x}
were non-degenerate, it would follow from Jensen's inequality
for strictly convex functions that
= e -ta{x}
(e -sa{x}) tIs
which is impossible.
obtain
n{x} = a{x}
Hence
a.s.
a , it follows that
n{x}
must be degenerate, and by (12) we
Writing
n:<: a'
a'
for the purely atomic part of
a.s., and since (10) and (11) may be written
in the form
Ee
(14 )
-t (n-a')U
= Ee-t(~-al)U
Ee-s(n-a')I :<: Ee-s(~-al)I
it is seen from Theorem 3.4 that
This shows that
that
Lj t;.nj ~
t;.
Ij~nj ~ ~
(n
(n
E
~
U
E
U ,
I
E
I ,
... a' d n - a' , i.e. that
, and since N'
Nil)
d
t;. = n
was arbitrary, even
, proving the first assertion.
E N)
The assertion involving regularity of
{~nj}
follows by the usual
arguments, so it remains to prove the assertion involving (9).
In this
case we get by Fatou's lemma
(15)
EnI
:S
liminf
miN"
L E~
j
.I
nJ
:S
limsup
n-+oo
L E~
j
.1 ~ aI
nJ
+
AnI
= E~I
I
E
I ,
-98replacing (11), so (13) is now replaced by
(16)
If
n{x}
were non-degenerate, we would obtain from (12), (16) and Jensen's
inequality
~
which is impossible, so again we get
that
~ - a'
NOTES.
n
~
a'
gn ~
a.s.
e
-ta{x} ,
= a{x}
n{x}
a.s.,
x
€
S , proving
Thus it follows by (14), (15) and .Theorem 1.4 that
a' , and the proof may be completed as before.
The canonical representation in Theorem 7.2 and Corollary 7.3,
which is a well-known classical result for random measures on
due in the present generality to Kingman (1967).
via Lemma 7.1
a~
R, is
The present approach
well as the convergence criterion in Theorem 7.2 were
given in Kallenberg (l973a).
Corollary 7.4 is a classical reSUlt, proved
'" "
in particular cases by Palm (1943), Hincin (1960) and Ososkov (1956), and
in general (on
R) by Grige1ionis (1963), (see also Goldman (1967) and
Jagers (1972)).
Finally, Theorem 7.5 is taken from Ka1lenberg (1973a)
while Theorem 7.8 is new.
PROBLEMS.
7.1.
Show in analogy with Lemma 6.6 that the random measure
7.2 is distributed like
process on
a
+
J xotn(dxdt)
R'+ x S with intensity
y
, where
n
~
in Theorem
is a Poisson
-99-
7.2.
{~nj}
Let
be a null-array of point processes on
S.
Show that
I.I;. is tight with only Poissonian limit distributions iff
J nJ
\.P{I;
.B > I} + 0 for every B
LJ
nJ.B > O} is bounded and L\.P{c
J snJ
7.3.
E
B .
State and prove a similar tightness criterion corresponding to the
convergence assertion in Theorem 7.2.
7.4.
Prove Corollary 7.4 directly from Theorem 4.8 and Lemma 6.1 in the
particular case when
7.5.
A
E
Md
Prove Theorem 7.8 in the point process case.
Show that
I may
be assumed to be a covering class even in the random measure case,
provided
7.6.
~ ~
Let
a
is known to be diffuse.
lCa,A)
on
S , let
C
c
B be a covering class and let
;t O} = 0
i f A is a-regular w.r.t. C
a
and that the converse is also true provided C is a DC-semiring
a
>
and
7.7.
Show that
0
E~ E
M
be a null-array of random measures on S such that
Show that A{ll * ;t O} = 0 i f {~nj } is
I.~ . §. some I (a, A)
E:
J nJ
for
every
a
>
E:
a-regular w.r.t. B~
, and that the converse is
Let
{~nj }
also true provided
8.
A{ll
E~ E
M.
COMPOUND AND THINNED POINT PROCESSES.
The notions of compounding and thinning were introduced in Section
1, and in Corollaries 3.2 and 3.5 we proved two uniqueness results for
compound point processes.
The aim of the present section is to study
the convergence in distribution of such random measures, and we shall
-100first consider a case when the limits are themselves compound point processes.
THEOREM 8. 1•
d
Sn
-+
E,
-+
d
n
no
For
n
d
E
nn
Bome
and that
E,
for some
B
some
C
E
with
p
= inf
80
nO
and
with
be a 8n -compound of nn . Then
d
Conversely" suppose that
~n -+ ~o
Pn > 0
~ a
Then
is tight.
{nn}
is a So-compound of
~
{Cn } is 2-reguZar w.r.t.
n
nO ' and if
l;C
thinning of nn' n
i: n
imply that
13 0 and
-+
Z+ ' let
there exist some constants PO' PI' ... E (0,1]
d
d
such that 13' -+ 8 ' and n'n -+ nO, " where ~ 1-S a Pn0
n
Z+ ' while
E
n
Let us first suppose that
be a 13 n -compound of
for
"
(1)
PROOF.
B~
nn'
n
E
sn
~
80
~~
and
E
Z
+
nO ' and let
By (1.5) and Theorem 4.5,
Z+
l;n
d
l;n -+
£'0
is then equivalent to
(2)
where
~n
E exp(nnlog
n
E
0
f)
-+
E exp(nOlog
~O
0
f E
f)
Fc
Z+ ' and by continuity and boundedness, (2) will
follow if we can prove that
f
(3)
By Theorem 5.5 in Billingsley (1968), it
random
(4)
nn' i.e. to show that
v
lln -+ 110
1S
E
F
c
enough to prove (3) for non-
implies
f
E
Fc
-101-
Considering a fixed
f
E
Fc ,we may assume that
]In
4.5), and after a suitable normalization that the
measures.
]In
;t
(cf. Problem
0
are probability
But then (4) may be written in the form
for some random elements
f
w
-+ ]JO
{log </>
is bounded, the sequence
n
0
in
S
with
f}
is uniformly bounded, and hence
Since
(5) is a consequence of the relation
(6)
Applying Theorem 5.S in Billingsley (1968) once more, it is seen that
the quantities
Tn
= sn
may be considered as non-random with
and since in that case
xn
= f(sn)
-+
whenever
that the convergence
</>n
N' c
-+
</>0
is uniform on bounded intervals.
sn ~ s
BE B
d
n B -+ 0 ,
n
~
so by (1.5) and bounded convergence
Ee
-s
B
n
=E
exp(n B log </> (1))
n
n
is tight.
and that
0 E N , we may assume that
d
N be such that nn -+ some n (n EN')
we obtain for any fixed
So '
But this follows from the fact
0 E M is a O-compound of
the sequence
-+
= x , it remains to prove that
f(s)
Let us now suppose conversely that
Since
sn
-+
1 ,
s
If
~ 0
n
Let
d
0
,
-102:and hence
Thus
n
F,B
~ 0,
B
E
~ = 0 a.s.
B , yielding the contradiction
~ 0 .
We shall use this fact to show that
{Sn}
is tight or, what amounts
to the same, that
lim limsup (1 - ~n(t)}
t-+O
n-+oo
(7)
=0
.
Suppose on the contrary that the limit in (7) is greater than some
Then there exist arbitrarily small numbers
sequence
t >
a
E
a .
>
such that, for some
N" eN' ,
(8)
1 -
Choosing
B
E
B~ n B
n
1 - x
(9)
~
~
n
(t) > E
such that
- log x
~
nB
n
E
N" .
~ 0 and noting that
2(1 - x)
it follows by (1. 5) and (8) that
Ee
-tl;n B
=
E exp(nn B log ~n(t))
I; B ~ I;B
n
~
E exp[-nnB(l - ~n(t)}]
~
Ee
-En B
n
d
-t!;B
-EnB
~ Ee
Letting
n.B , we obtain Ee
··
1 ~ Be-En. B < 1 , provIng
.
t -+ A , we reac h th e contrad lctl0n
t h at (7)·IS
and since
indeed true.
and
~B-+
Thus the sequences
{Sn}
and
{nn}
are both tight, and
in particular we may consider convergent subsequences and use the direct
part of the theorem to conclude that
Let us now suppose that
C
E
{C~}
I;
has the asserted form.
is 2-regular w.r.t.
B~'-> with I;C ~ 0 • and assume that Sn ~ S'
and
n
n
BI;
i
n'
for some
as
n -+
00
-103through
N'
N
S' ~ 0
it follows from (1. 5) that
c
Since
E;
is then as' -compound of
simple (cf. Problem 4.9) .
Put
of
R+~valued
n'
and let
So
(10)
be an
P{SO
Then
= LS
¢0
> x} =
and
so letting
o
f
E
P{S'
cPO
nO
F,;
O} , let
>
must be
Cn'
be a p-tFtinning
nO
random variable satisfying
> x}
satisfy the relation
0
f)
P<P
cPt ::
O
+ (1 - p)
,
and using (1. 5) and (1.6), we
nO
L , (-log[l
n
=
= Ln ,T-log[l - (1 - <P'
and hence
and so
F
L (-log
=
-1
be a So -compound of
E;O
obtain for any
LF,; (f)
p
= LS '
cP '
0
=
P
= BE;
B
n'
P{S'
, and moreover
n'
pel - <Po
f)]) = L ,(-log <p'
0
0
n
o g F,; by Theorem 3.1. Now P{So
O}
>
=
f)J)
0
=1
f)
= Lt"(f)
,
~
, and moreover
is simple by Problem 2.4, so it follows from Corollary 3.5 that
are uni~uely determined by
and
y
Choose any fixed
and define
liminf p
n n
p
> ()
n
=
P{s
n
>
y} ,
since i f Pn
N8
satisfying
> 0
n
-+
0
E
n
-+
we could choose some further subsequence
and then (10) would hold with
0
Then
Z+
as
pF,;-l
>
PO
y}
>
> 0
0
Ns O = y} = 0 ,
and
, and moreover
through some sequence
ClO
8
n
such that
N'
d
=
P{S
n
> y} -+
P{S' >
y} =
some
Nil
c
N'
p = P{8' > O}
>
0 , and we would get the
-+
contradiction
Pn
c
P{B'
>
O}P{SO
> y} >
0
(n
E
Nil) .
N ,
S' ,
-104-
p
= inf
the sequence
{Pn}
To obtain
pn > 0 , we may replace the finitely many zeros in
by arbitrary positive numbers.
be a Pn-thinning of
nn'
n
We can now let
and define random variables
E Z+ '
~
8n'
satisfying (1).
N' eN, we may choose a subsequence Nil c N'
Given any sequence
d
8 + some
n
such that
In this case the constant
P{13'
>
O}
that, on
=
Nil
-1
n'n
'I
d
+
was shown above that
like
p
and
no ' a p'-thinning of
n
is a
n'o a po-thinning of
> x}
n'
in (10) equals
p
n
-1
p'
+
p'PO
= pp' -1 P{I3'
> x}
n' .
= P{ 13'
o~
13"
13'0
(n
E
n , and we get
(n
d
n'n
+
n' , and so
n'0
(n
E
is
Furthermore,
Nil) , where
E
= pp' -1 p'Po-1 P{8 0
13'n ~ 8'0
and it follows that
Nil) .
nO
> x}
= PPO-1 P{13 0
(n
Nil) .
E
> x}
=
x}
In view of Theorem
2.3 in Billingsley (1968), this completes the proof.
We are now going to consider the entirely different case when the
compounding variables
I3 n
satisfy
8
n
d
+
O.
The class of possible limits
may then be described as follows.
Consider arbitrary
a
1/J (t) =
at
R
and
+ A(1
_ e
E
+
with
A E M(R' )
+
-IT
(11)
Nil) .
,and in particular
= N13 0 >
so
> y}
On the other hand, it
(p' po 1)-thinning of
13'n ~ 8"0
it is seen from (1) that
P{I3H
some
+
Using the direct part of the theorem, it is seen
p'PO
,
d
nn
13' ,
t)
t
E
R
+
AK <
00
,
and define
-105-
By Theorem 7.2, there exists for every Y
with homogeneous (w.r.t.
any
f
E
E
M some
random_measure
~
y) independent increments satisfring, for
F ,
log Ee -~f
= oJ.lf
+
J
=
(l - e -Jef (t)) (A x )l) (dJedt)
R'xS
t
=
+
{af(t)
+
J
R
By Lemma 1.4, we may consider
,
e-xf(t))A(dx)}y(dt)
(1 -
= )l(t/J
0
f) .
+
)l
=n
as a random measure and mix w.r.t.
its distribution, thus obtaining
(12)
L (t/J
0
n
Note in particular that, if
directed by
n.
a
=0
and
f)
f
E
A = 01 ' then
~
F .
is a Cox process
We shall need the following uniqueness result which is
similar to Corollary 3.5.
LEMMA 8.2.
for some
~
Let
n
~
a
diffuse for some
be a random measure on
and
C
E
uniqueZy determined by
PROOF.
S
A ~ and suppose that
B wi th
nC ~ 0 •
Then
a
satisfying (11) and (12)
AK
+
Pn
-1
=1
~
a
whiZe
and
Cn
is
A are
p~-l.
By (12),
B
and replacing
B by
E
B ,
B n C , it follows by Theorem 3.4 that
is uniquely determined by
p~-l.
P(Cn)-l
But then it follows as in the proof
-106-
of Corollary 3.5 that even
~
determines
A .
and
a.
is uniquely determined, and by Lemma 6.1,
~
The uniqueness of
Pn
-1
now follows as in
case of Corollary 3.5.
To avoid trivial complications, let us assume that any compounding
variable
is such that
B
f3
d
~
0 , and further that the
(11) are normalized in such a way that
THEOREM 8.3.
For
n
E
N
~
let
Suppose that
i
=1
Then
~n
5!
in
.
~
be a Bn-compound of
0
A
"'f'+'
v J
0
and put
c n +d 0
n n
the conditions
Purthermore~
d
cn nn
(i)
B
n
~n
a. + AK
and
a.
+
n ,
(ii)
imply that
~
d
~
implies that
if moreover
then
Pn
PROOF.
-1
~
Now
large
a.
Writing
Ee
B
n
~
B
n
n
satisfies (11) and (12) for some
=
and
~
n
N
ConverseZy~
satisfying (11) and (12).
B~
n
for some
~
a.
CE
and
B~
A ~ and
with
are unique and satisfy (i) and (ii).
A
= LB
n EN, we get as before
n
E exp(nnB log ~n(t))
implies that
0
E
~
some
{c n Cn n } is regular w.r.t.
-t~
(13)
+
n
~
n
+
BE B
n
EN.
1 , so by (9) we get for sufficiently
~C ~
0 ,
-1072c
n
and it follows from (13) that
E exp(-2cnnnB) ~ Ee
showing that
r~n B
~
0
iff
B
-~
n
cn nn B
~ E exp(-cnnn B)
d
+
B
0 •
E
B
E
B ,
This proves the first
B •
assertion.
Next suppose that (i) and (ii) are satisfied.
proof of Theorem 8.1, we obtain
Proceeding as in the
~ n ~ ~ with p~-l determined by (11)
and (12), provided we can show that
whenever
x
n
+
x .
But this follows from the
-1
fact, essentially being proved in Lemma 6.1, that
-c n log
~n
tends to
W uniformly on bounded intervals.
Suppose conversely that
Then
!;
n
B ~ ~B
~n ~ some ~, and let B E B~ be arbitrary.
by Lemma 4.3, so by (13)
and since this limit tends to one as
some
(14)
t
E
(0,1)
t
+
0 , there exists for each
satisfying
E exp(n n B log ~ n (t)) ~ 1 - E/2
n
EN.
Now it follows from (9) and the elementary inequality
E >
0
-1081 - e
-tx
~
x
tel - e- )
t
[0,1]
E
that
-tS
- log ~ (t) ~ 1 - ~ (t)
n
=
n
E(l _ e
n} ~ tE (1 _ e
-6
n) = tc
n
'"
and so by (14) and Cebysev's
inequality
'"
P{ cn nn B > t
-1
log 2}
=
P{tcn nn B > log 2}
=
PO - exp(n B log ~ (t)) > !.}
n
n
2
~
2EO - exp(nn B log ~ n (t))} ~
proving tightness of
by Lemma 4.6 that
Let us
no\'l
Since
{c 11 B}
n n
B
~
P{-nn B log
~
E B~
n (t)
>
log 2}
~
£
,
was arbitrary, it follows
is tight.
{c n }
n n
~
assume in addition that
2 o.
Proceeding as in the
proof of Theorem 8.1, we can then prove that
lim limsup c-1(1 - ~ (t)) =
n+oo
n
n
(15)
V
t~O
a .
\I
By Cebysev's inequality, we further obtain for any
-S
r}
>
and letting in order
n
~
r
~
00
lim limsup c-lp{S
n~oo
which shows that the sequence
R
+
is tight.
a
t
c-lE(l-e n) c-l(l_~ (t)}
_n
= _n
n
_
rt
rt
l_el_e-
+00
r~oo
r,t >
n
and
t
>
r}
n
{c-lK(pS- l )}
n
n
~
a ,
we get by (15)
=a ,
of probability measures on
In particular there exist some sequence
N'
~
N and some
-109n,
A such that (i) and (ii) hold on
and
a
Nl , and we may conclude
~
from the sufficiency part of the theorem that
has the asserted form.
~ ~ 0 we may take n = 0 and choose arbitrary
(In the case
and
a
A .)
is regular
Let us finally assume that
B~ for some C
w.r.t.
{c-lK(PS- l )}
and
n
a
and
B~ =
N"
c
N'
2 O.
Since the sequences
~
Cn
n
may thus conclude from Lemma 8.2 that
p~
-1
{cnn n }
N must contain a
c
N"
n,
for some
satisfies (11) and (12), and in particular
B ' so it follows by Problem 4.9 that
determined by
N'
such that (i) and (ii) hold on
But then
A
B~ with ~C
are both tight, any sequence
n
subsequence
E
Pn
-1
is a.s. diffuse.
a
»
and
A
We
are uniquely
,and hence, by Theorem 2.3 in Billingsley (1968),
that (i) and (ii) remain true for the original sequence.
This completes
the proof.
The above convergence criterion takes a particularly simple form
in the case of thinnings:
COROLLARY 8.4.
For
of some point process
iff Pn nn
PROOF.
i
some
n EN, let
Pn E (0,1]
Suppose that
n, and in this case
The preceding proof applies with
Pn
~
and let
~
0 •
be a p -thinning
n
n
Then
~
d
n
~
some
~ is a Cox process directed by n.
c n - Pn '
a
=
0
and
A
=
6
1
(and it may even be simplified in the present case, since eii) is automatically satisfied).
of Corollary 3.2.
No regularity assumption is required on account
-110-
The last result leads to an interesting characterization of the class
of Cox processes:
COROLLARY 8.5.
~
Let
be a point process on S.
process d-&l'ected by some random measure
~.
a p-thinning of some point process
directed by
P
PROOF.
p
Let
and
11
P
-1
-1
E
(0,1]
, let
and
~
L~ (f) =
P
L
11
=L
(-log[l
f
pel - e- )])
p
f
-1 (p(l - e- ))
P n
so
p-thinning of some
,
np
p
E
(0, IJ
is a Cox process
=
be Cox processes directed by
be a p-thinning of
~p
f
By
np
F
E
f
L -1 (1 - [1 - pel - e- )])
p n
f
L (1 - e- ) = L~(f) ,
n
~
=
Conversely, suppose that
~n = ~
is for every
In this case
np
and (1. 6) we then obtain for any
(1. 4)
~
is a Cox
n.
respectively, and let
11
n iff
~
Then
is for every p
~
Applying Corollary 8.4 with
n EN, it is seen that
~
p
n
=
E (0,1]
a
-1
= nand
must be a Cox process.
We conclude this section with a useful criterion for infinite divisibility.
LEMMA 8.6.
11
C
iff
Let
~
be a random measure on
c~.
be a Cox process directed by
11
PROOF.
c
Then
is infinitely divisible for every
The necessity part is obvious.
infinitely divisible for every
c
>
O.
and, for any
S
~
c > 0 , let
is infinitely divisible
c > 0 .
Suppose conversely that
Put
n
c
p~-l = D and introduce
is
-111-
in the spaces of distributions on
and
J
corresponding to division by
Cox processes respectively,
that
V
p
and
J
t~e
np
V
P
p , p-thinning and formation of
M
Nand
p E: (O,lJ .
operators
From (1.4) and (1.6) it is seen
commute with convolution.
Using this fact and the last
assertion in Corollary 8.S, we obtain
(V (In D)l/n)n
P
Thus
(JD)l/n
P
= V In
P
P
D
= JD
is a p-thinning for every
Corollary 8.5 there exists for every
M such that
P E: (0,1]
UD) lIn
= JDn
p E: (0,1]
n
and
EN.
nE: N , so by
n E: N some distribution
Dn
on
But then
n E: N ,
and it follows by Corollary 3.2 that
o
D = (D )n
n
n E: N , proving that
is infinitely divisible.
NOTES.
Theorem 8.1 is new while Theorem 8.3 and Corollary 8.4 were given
in Kallenberg (1975).
The latter result extends some classical work of
Renyi (1956), Nawrotzki (1962), Be1yaev (1963) and GOldman (1967), who
all consider p-thinnings of some fixed point process and change the "time"
scale by the factor
p-1
to be able to obtain convergence.
Finally,
Corollary 8.5 is due with a direct proof to Mecke (1968 ) and (1972). (ef.
Kerstan, Matthes and Mecke (1974), page 318), while Lemma 8.6 is due to
Kummer and Matthes (1970).
-112-
PROBLEMS.
8.1.
State and prove the thinning case version of Theorem 8.1.
Note
that no regularity condition is required in this case.
8.2.
Let
be a 8..f,Qmpound of s.ome simple point process
f,
p '" P{B > O}
*
Show that
n and put
is then a p-thinning of
E;O+
n.
Use
this fact to give a new proof of Corollary 3.5.
8.3.
8.4.
N , let E;
be a B -compound of
n
n
d
B ~ some 8 2 0
Show that E; + some
n
n
and that E;, is then a B-compound of n
responding version of Theorem 8.3.
For
n
E
nn , and suppose that
d
E;,
iff nn + some n ,
State and prove the cor-
Let E;, be a B-compound of n and suppose that 8 is infinitely
divisible. Show that E;, has then infinitely many essentially different representations as a 8'-compound of
(Hint: Replace
n by nn,
n
E
Theorem 8.1 to reduce to the case
N
n' with 8'
>
0
a.s.
Proceed as in the proof of
8 > 0
a.s.
Finally prove
that the representations thus obtained are really different.)
8.5.
Let
E;,
be a 8-compound of
divisible.
Show that
E;
n and suppose that
n
is infinitely
is then also infinitely divisible.
Prove
the corresponding result for the random measures defined by (11)
and (12).
8.6.
Let E;, be a Cox process directed by n or a p-thinning of n
Show that it is possible that E;, is infinitely divisible while
n is not. (Hint: It is enough to consider Z+-va1ued random variables.
To construct our
n , we may start with the familiar
Raikov-L~vy
example reproduced in Lukacs (1970), page 251, and add a suitable
infinitely divisible variable to make the thinning or Cox variable
E;,
infinitely divisible.)
·e
-113-
9.
.-
SYMMETRICALLY DISTRIBUTED RANDOM MEASURES.
Let
w
M
d
E
be fixed.
We shall say that a random measure
symmetricaZZy distributed with respect to
w
s
is
if the distribution of
kEN , only depends
If
S
with distribution
oT ,
process
wS
E
R'+
and if
is a random element in
T
w/wS, then this property is obvious for the point
and since the property is persistent under addition of in-
dependent random measures and under mixing, it is shared by all mixed
In particular,
Poisson and sample processes which are defined w.r.t. w
any such process must satisfy
(1)
HwB)
P{sB = O}
for some non-increasing function
w.r.t.
P{sB = O} = Ee
so in this case
¢ =
-awB
P{~B
(4 )
1/J
B ,
is a mixed Poisson process
a , then
¢(wB)
B
E
On the other hand, if
La
is a mixed sample process w.r.t.
(3)
=
E
s
If
<p.
w with mixing random variable
(2)
where
B
_
- ~~)
O} = F. (I - wB)'V
wS
- 1/J (1
wS
1/J(1 - w~) ,
1/J(t)
=
<t>(wS(l - t))
E
R'
and i f
+
t
= <PewB)
B
E
B
'V , so in this case
is the generating function of
=
wS
E:
[0, wS)
t
E
(0, I] .
s
'V , then
with mixing variable
w/wS
<Pet)
B ,
,
-114rf
~
is a mixed Poisson or sample process, then
to bc dctermined by
denote such a process
a
>
We shall use the symbol
(Note, however, that the pair
~.
uniquely determined by
for any
~.
w and
p~-l is easily seen
p~-l since it can be replaced by
M(w,~)
(w,~)
to
is not
(aw, ~('/a))
0 .)
It turns out that every symmetrically distributed simple point process
is a mixed Poisson or sample process.
We shall even be able to prove the
following stronger result.
THEOREM 9.1. Let
~
be a simpZe point process on
S and Zet
Further suppose that
U c B is a DC-ring and that
t
d
Then
~t
~ =
some
iff there
M(w,~)
ex~st8
E
(O,~]
wE M
d .
is fixed.
some non-increasing function
satisfying
(5 )
Ee
-t~U
= ~t CwU)
UE U.
In that case
(5' )
PROOF.
X E
Let
t
E
R'
+
and
B
process with mixing variable
E
B
a.
[0, wS] .
and suppose that
~
is a mixed Poisson
Then
On the other hand, we get in the mixed sample case, on account of (4)
-115-
.Thus (5) holds in both cases with
qit
defined by (5').
For
t",
00
this statement was proved in (2) and (3).
Suppose conversely that (5) holds, and let us first assume that
t
=
w
~
w
=0
Since
0
Our first aim is to extend (5) to
this purpose, let
clearly implies
some sequence of sets
0
<
wU k
~
=0
a.s., we may assume that
B, i.e. to prove (1).
For
sl' 52' ... E S be a bounded sequence of distinct
points belonging to the support of
Then
~
00
w{s}
w, and consider for fixed
Uk E U with
=0
while
s
= sn
kEN ,satisfying
~Uk ~ ~{s}
~
Uk
Is} .
a.s., so we get
Hence by Fatou's lemma
so
qi(O+)
on
[0, wS)
=
1.
To extend this continuity at
(or even on
a
be fixed and choose
with
s
<
t
<
s
+ cS ,
s -
(6 )
In fact, let
C
E
>
[0, wSJ
0
if
such that
wU
S S <
U be such that
t
wC
<
to (uniform) continuity
w has bounded support), let
£
>
1 - qi(30) <
E
[0, wS)
we can choose sets
cS <
0
lJ,V
wV
~ t
<
U with
E
t
For any
£.
+
s,t
0
U c V and such that
0 .
and let
{C .} cUbe a null-array
nJ
-116-
max wC . + 0 since w E M ' so it suffices
d
j
nJ
for construction of U and V to consider an"n E N' with max wC . < Q
j
nJ
From (5), (6) and the monotonicity of ~ we get
of partitions of
~(s) -
~(t)
C
~ ~(wU)
Then
-
~(wV) = P{~U =
O} -
P{~V =
~ P{~ (V\U) > O}
=
=1
O} =
P{~U =
n
1 - ~(wV - wU) ~ 1 - ~(3Q) <
Nand
E
h
<
~V >
O}
~
- ~ (w(V\U)) =
E
We may now use the monotone class theorem to extend (5) to
For arbitTary
0 ,
•
B.
wS/n , it is possible to choose disjoint
Bl , ... ,Bn E B with wB = ••• = wB = ~
It. fact, let C E B
I
n
with wC > nh and let {C .} be a null-array of nested partitions of
mJ
-1
< k
For fixed k > 0 we can choose m so large that max wC
C
mj
j
and then there exist disjoint unions Bkl , ... ,Bkn of the partitioning
-1
~ h ,
< wB
j = 1, ... ,n
sets CmI' Cm2' ... such that h - k
kj
sets
The sets
Bkj
may clearly be choosen non-decreasjng in' k
j , and in that case the sets
B. = U B .
J
k
kJ
j
= 1,
for each
... ,n , have the
desired property.
With this choice of
B , ... ,B n ' it is seen from (1) and Lemma
l
5.7 that
?:
whichSmcans that the function
pp. 223,439).
5
If
wS
= ~ ,
~
0 ,
is completely monotone (cf. Feller (1971),
it follows that
is the L-transform of some
together with the corresponding inequality for arbitrary
x
-e
-117-
R+ -valued random variable, and we. get
On the other hand, if
3.3.
t
E
wS
<
by (2) and Theorem
~(t)
then the function
ro
= ~(wS(l
- t))
(0,1) , being absolutely monotone, is the probability generating function
of some Z -valued random variable, and again
Theorem 3.3.
= M(w,~)
This completes the proof in the case
In the case of finite
p = 1 - e
d
~
+
-t
t , let
t
, now by (3) and
= ro
be a p-thinning of
~
~,where
As in the proof of Theorem 3.4, we obtain
UE U,
and it follows as above that
distributed w.r.t.
d
n = M(w, ~t) .
n
~.
In particular, (1) must hold
it follows as above that
~,and
is symmetrically
w, and from (1.6) it is easily seen by analytic con-
tinuation that this is also true for
for some
But then
The proof is
now complete.
COROLLARY 9.2.
~ d
=
then take
to
-:f.'+'
v J
M( w,~ )
some
[0, wB]
w
B
.
for any fixed
~
Let
=
be a point process on
= some
B~ d
M( w '
B
Bw , in which case
Furthermore,
B
E
B with
p~-l
wB >
~B
.)
~B
S and let
f or every
B E
w EM.
B
.
We may
becomes the restriction of
is then uniquely determined by
Then
~
P(t,;B)-l
°
Note that the only if part of the first assertion contains a wellknown classical property of the Poisson process.
PROOF.
Let us first suppose that
w
E
Md'
Then the first assertion
is an immediate consequence of Theorem 9.2, while the second one follows
-118-
by comparison of (Z) and (3).
~B
As for the last assertion, note that
has probability generating function
~(wB(l - t)},
P(~B)-l determines
and hence, by the uniqueness of analytic
continuations, on
~ on
[0, wB]
t
E
[O,lJ
Thus
[0, wS)
The proof for general
w
E
M requires randomization.
Write
~ =
L. is T.
J
and choose independently of
a sequence {n.}
J
J
variables which are uniformly distributed on [0,1]
seen that
~ =
L. is (T.,n.)
J
J
of independent random
{T.}
is a point process on
J
It is then easily
S x [0,1]
and that
J
Add
~ = M(w x A,~)
iff ~ =
M(w,~)
A denotes Lebesgue measure on
, where
,..
[0,1]
Since
so it holds for
w x A is diffuse, the assertion is true for
~,
and
as well.
~
We next indicate by an example how Theorem 9.1 can be used to improve
Theorems 3.3 and 3.4 in the particular case of Poisson and sample processes.
COROLLARY 9.3.
Let
w
E
Md
and~
for
wS =
or
00
<
00
Poisson or sample process respectively with intensity
that
for some DC-ring
= O} = ~(wU)
~
be a
Further suppose
w
UE U,
U c B and some non-increasing function
d
= n iff theY'e exist two sets Bl , Bz
(7)
let
n is a simple point process on S satisfying
P{nU
E;,
~
P{~B.
J
= O} = P{nB. = O}
J
E
B with 0
<
= 1,2
.
j
wB l
~
<
ThBn
wB Z satisfying
-119-.
PROOF.
d
By Theorem 9.1 we have
n =
.
In the case
M(w,~)
wS
=
we
00
get by (2) and (7)
Ee
(8)
s = wB l
where
~.
If
and
-as
= e -s
Ee
t = wB 2 while
a
-at
= e -t
is a random variable with L-transform
were non-degenerate, it would follow by Jensen's inequality for
a
strictly convex functions that
Ee -at
.
which is impossible and proves that
we get
a = 1
a. s., and so
process with intensity
~ (x)
w.
==
a
e
= e -t
is indeed degenerate.
-x
which means that
n
From (8)
is a Poisson
A similar proof applies to the case
wS
<
00
•
We are now going to characterize the class of arbitrary symmetrically
distributed random measures.
measure
with homogeneous w.r.t.
~
- log Ee -H
(9)
Consider an infinitely divisible random
=
awf
+
f
w independent increments such that
(1 - e-xf(s))(w x A) (dsdx)
f E
F,
SxR'
+
where
a
integral
R
E
+
and
Is (1 -
A E M(R')
with
+
e-xf(s)}w(ds)
AK
<
00.
By Fubini's theorem, the
is a measurable function of
x, and
therefore the right-hand side of (9) is jointly measurable in
A
A
Hence, by Lemma 1.4, we may perform mixing with respect to
considered as random elements.
in case
wS
=
00.
The case
wS
<
a
and
a
and
This gives the most general distribution
00
is entirely different:
-120-
Let
THEOREM 9.4.
~
Then
~
and let
Md\{O}
W E
be a random measure on
is symmetrically distributed w.r.t.
in the case
(i)
wS =
00
~
that~
given
A on
with
R'
+
AK
<
a.s. such
00
A ~ the conditional L-transfoY'm of
and
a
w iff
there exist some R+-valued random variable
and some random measure
a
S .
~
is given by (9);
(ii)
~n
the case
a
and some point process
wS
<
00
,
there exist some R+-valued random variable
S
= L'Oo
] ..,.
on
+
]
a.s. such
that~
as
L.S.o
] ] T.
aw'
+
given
~
a
where
and
WI
S
independent random oZemp.nts in
measurable functions of
We shall call
elements of
PROOF.
(a,A)
~
(a,A)
and
(a,S)
or
(a,S)
00
are
T
WI
respectively are a.s. unique
respectively the canonical random
~.
~
is symmetrically distributed w.r.t.
that
wS
<
,
say that
=1
wS
B
by monotone convergence to
(10)
<
.
that
a.s.
I , T 2.
S with distribution
while
The sufficiency of (i) and (ii) is obvious.
00
L.S.
] ]
is conditionally distributed
~
= w/wS
J
The random elements
~
with
RI
Hence by Lemma 2.1,
f,
Conversely, suppose
w, and let us first assume
The symmetry assumption then extends
and in particular it it seen that
has the representation
I;;S
<
00
-121for some random elements
sd
8 1 ~ 82 ~ ... ~ 0
Md ,
E
the latter being a.s. distinct.
Whenever two or more
,.
assume that the corresponding
w is Lebesgue measure.
Ps-l
and
1 - 1 mapping
8 , 8 , ...
1
2
distributions of
S ,given
1
measurable function of
so even
Ps
l
d
=
S
J
is the unit interval
sT- l ~ s
T of
[0,1)
for every w-preserving
onto itself.
[0,1)
Since
a
= sdS
(a, 81 , 82 , ... ) , are again symmetric, so we
as constants.
Since
S , we have for any
is sYmmetric.
[(j - l)/n, j/n)
T
sd
is by Lemma 2.1 a
as above
Moreover, the process
-1
sdT
=
(sT
sd[O,t) - at,
and
'nnj
= Sd1nj - o./n,
max
.
In.
TIJ I ~
j
= 1,
... ,n,
we get
I.
J
2
Tl n J'
~ 0:. ITl nJ. I)
J
J
°
a. s. ,
n~oo.
From the symmetry and the obvious relations
and
we hence obtain by bounded convergence
=
E
I
j
2
Tl nj ~
°,
and so
~x
<
coincide, we may
B.
-1
t
d
)d = sd '
E
[0,1]
is clearly a.S. uniformly continuous and of bounded variation so, writing
I nj
S ,
are invariant under such transformations, the conditional
a, 6 , 82 , ...
may consider
E
Using Theorem 3.1, it is easily seen that
is symmetric in the sense that
bi-measurable
'1' '2' '"
are taken in random order.
J
We shall first consider the case when
while
and
K
E{jIl nnj}
2
= m~X{kEn~l
~ nETl~l
+
+
k~
- llEnn1
"n2} ,
nen - 1) !ETl nl Tl n 2 1
~ 0 .
n EN,
4It
-122~d[O,t)
But this implies
and therefore
t;d = etA
E
of
by
N
'1
6 1 > Q , and let
nv
which maps
[0,1)
t;T- l
m
t;nm
I
E
a.s. for every rational
For
k,m
Ink
onto
E
E
A}
I n,k+m
L
A,
E
= k + m} =
\/
A
= L
k=l
P{t;T~
= P{t;T-\1m1
E
n
E
A, \/
CO = k} =
L
k=l
A} = P{I;
not uniquely determined by
nand
nm
E
t;
that
min
s
snm[O,l) =
d
=!;
d
nm
~
A,
E
P{I;T- l
\/m
\/(t;T
-1
km
) = k + m} =
A, v(1;) = k} =
E
A} .
81 = 82 ' although
in this case.)
we now define
(0,1]
by
'1
is
v
This proves that
I
t;[0,1) =
.
t; s , l.e.
measurable function
with
s
I;s[O,l)
t;s
~
t;
t;s
d
t;nm = I;
as the random measure obtained
Let
'1 + s (mod 1)
s , and suppose that
for any interval
S
E
by replacing
~
t;
Ink ' and put
m.
For fixed
from
onto
M
L P{~T~
(Note that this relation is even true when
for any
E
I n,k+m
k=l
k=l
n
be defined for fixed
n
P{t;
{O,l] ,
E
, let Tkm be the transformation
and
By symmetry we get for any
=
= v (~)
v
Z (mod n)
n
P{t;
t
a.s.
Next suppose that
n
= at
m,n
'1 + s ~ 0 (mod 1)
t;nm
w
~
t;s
00
Then
¢ I , and hence also when s
Therefore
~
E
in such a way
!; I
nm
0
1
~
t; I
s
since
a. s. , and we get
By Fubini's theorem, this implies for any
k
f: R+ ~ R+ '
k = sup{j: 8.
J
>
O} ,
-123-
'2' ... ) = Ef('1
Ef(,1'
f:
= E
, 1'
where
0") =
f (, 1 + s, '2' ..
is a random variable in
with distribution
w.
Hence
inductively, it is seen that
u).
'2'
+ s,
l
,
ds
=E
T
2
B.
~
Sand
parti tions of
of
w •
of
[0,1]
S
is compact.
0
)
ds
=
which is independent of
= [0,1)
S
w with
wS
= 1 ,
note first
extends by monotone convergence from
For convenience, we may compactify
beginning that
f (s, '2' ..
are independent with distribution
,
This completes the proof of (ii) when
that the sYmmetry property of
f:
'2' .. o)ds =
has the same properties, and continuing
Turning to the case of general
to
)
Ef('1 + s,
rO,l)
'I
T
0
f:
Let
B
S or rather assume from the
S = {S .}
nJ
c
B be a null-array of nested
S , and note that ~x wSnj+O by compactness of S and diffuseness
To each set
satisfying
S .
nJ
J
we can make correspond a subinterval
I I nJ·1 = wSnJ.
a null-array of nested partitions of
I
in such a way that
I
= {I nJ.}
For each
S
E
[0,1]
.
nJ
forms
S there is
S.
containing s , and hence we
nJ ns
can define a function f: S + [0,1] by res) = n 1-.
Each
n nJ ns
t c [0,1] , may be written as a countable union of S-sets, so f is measurable,
a unique decreasing sequence of sets
and it follows in particular that every measure
on
[0,1]
M , so
E,
Moreover, the mapping
induces a random measure
~ E
M induces a measure
~f
is measurable by definition of
[0,1] .
-1
-124-
Writing
of
D for the set of I-interval endpoints, we get by construction
f
t
and hence for any
t / wf
-1
[O,t]
wf
-1
measure.
wf
-1
wf
[O,s),!
=t
-1
f
[0 ,1]
tED ,
D, s
t}
>
~
inf{s
D is dense in
=1
implies that
D: s
E
t}
>
on
wf
-1
equals Lebesgue
is symmetrically distributed with respect to
S, it is easily seen that
f(S)\D , and since
w.
f
has a unique
and therefore
a.s., there is a.s. a unique correspondence between the atoms of
~f
-1
~
and
, and (10) is equivalent to
with a.s. diffuse
~ d f- I
and distinct
of the theorem is known to be true for
with
t ,
3
[0,1] , this together
is symmetrically distributed with respect to
~
-1
-1
Since
Using the completeness of
inverse
~
rO, 1] '" wS
~f-1
Moreover,
since
-1
[0, t]
with the fact that
wf
D\{l}
E
inf{wf
=
which yields
t
::;
= ~dS
a
, while
f(T 1),f(T ), .. ,
2
independent of
(a,6 ,6 , ... )
1 2
every
we obtain a.s.
S .
nJ
~dS
nJ.
E
=
S
t;d(S . \f
nJ
-1
D)
f(T ),f(T ), ...
2
l
~f
-1
, and so
Now the statement
-1-1
~df,
=
awf
are mutually independent and
with common distribution
-1
\D)
= t;df -1 (1 nJ. \D) = awf (1.
nJ
wf
=
= a I1nJ. I = awSnJ.
-1
awf
-1
.
For
1 . =
nJ
a.s.
-125~d
S being a OC-semi-ring, it follows by Oynkin's theorem that
and
a.s.
Similarly, we have a.s. for any
{'k
S .} ~ {'k
nJ
E
n,
S .\f-lO} = {f('k)
nJ
€
j
and
k
I .\O} = {f('k)
nJ
€
= aw
I .} ,
nJ
E
so we get
k
P({(a, Sl' SZ' ... )
A}
E
for any measurable set
k
(')
j=l
h.
E
J
= P{(a, Sl' SZ' ... ) E
B.})
J
A and for arbitrary
kEN
and by Oynkin's theorem, this extends to general
proof of (ii) is now complete.
CO
n
with
S
t
and
with respect to
wC l
distribution
0.
Ij
1 ' Sn
that
0.
m :::' n
E
1 = aZ
N
0
Pmn
Snj
= ...
Since
Pmn
by Corollary 8.5 that
hy some
A
m
we may take
, and that
A
n
+
Sm
C , CZ'
l
...
B
E
is synnnetrically distributed
'nl' 'nZ' ... , the latter being
and mutually independent with common
Cm~ = Cm(C~)
n
/wC
= wCmn
a
is obvious.
nj 0 T .
nJ
~
J
m
as
n
$
for
m
~
n , so writing
it is seen by identification
n EN
Sm is a p mn -thinning of
a.s. while
= a
(a,S)
+ \ S
and some
Now
The
, we know that
N
(a n , Sn l' Sn Z' ... )
Cn w/wC n
B ,
l
Choose
<Xl
Cn~
n E
= a n (Cnw)
an' Snl' SnZ' ...
independent of
a =
Since
Cn w for each
Cn~
for some
a
>
wS =
wB.
J
IT
j=l
and. B ,
l
The measurability of
We now consider the case when
A}
+
<Xl
for each fixed
m
E
N
is distributed as a Cox process on
d
A = Pmnn
A
m
= (wC ) A for each
n
Putting
n
Let
A
f
E:
Sn
for
, it follows
R'
+
directed
= A/weI , it is seen that
F
c
be arbitrary, and
-126-
choose
n
N so large that the support of
E
f
is contained in
Cn
Then
Ee -(~-aw)f
=
=
\
Eexp{-
L
j
I
S .f(T .)} = EE[exp{- L S .f(T .)}
nJ
nJ
j nJ
nJ
I
E IT. E[exp{-SnJ·f(TnJ·)}
J
SnJ']
=
E
{S .}]
nJ
=
fSexp(-s nJ.f(s))w(ds)/wCn
IT
j
and i f we define
hex)
=-
log
JS
e
-xf (s)
w(ds)/wC
n
this expression may be written
E
exp{-h(S .)}= Eexp{- L h(S .)}
.
nJ
.
nJ
IT
J
=
Ee
-S h
n
=
EE[e
-S h
n
I
1<] =
J
= Eexp{-(wC
n
)
fR' [1 - fSe-xf(s)w(ds)/wCn]I«dx)} =
+
fR'xS (1
= Eexp{ -
e -xf(s)) (I<
x
w)(dxds)} .
+'
Putting
f(s)
=
tIc (s) •
t
E
1
monotone convergence that
lim
R+ • it follows by Fubini's theorem and
L~C
t+O
a.s.
This proves that
in (i).
But for the
~
~
= 1
can only be true if
I<K <
00
is distributed as the random measure described
in (i) we easily obtain
strong law of large numbers, so
the
(t)
1
I<
S /wC
n
n
X I< a.s. by the
is a measurable function of
under consideration we may therefore redefine
I<
~
according to
For
•
-127-
this mapping, and doing so, (i) will automatically be satisfied.
This
completes the proof of the theorem.
Of particular interest is the class of random measures which are
at the same time sYmmetrically distributed and infinitely divisible.
Here is a simple characterization of this class:
THEOREM 9.5.
Let the random measure
with respect to some
or
(a, B)
or
(a,B)
PROOF.
be symmetrically distributed
Md with canonical random elements
W E
~
Then
~
(a,A)
is infinitely divisible iff this is true for
(a,A)
respectively.
First suppose that
For each
n
E
wS <
00
,
and let
(a,S)
be infinitely divisible.
N we may choose independent random elements
such that
By randomization we may then construct independent sYmmetrically distributed
random measures
From Theorem 9.4 it is seen that
(a , B ) •
n
with canonical random elements
~
'"~ l' ... , "'n
n
+
t;,
n
distributed with canonical random element
d
=
(a, B) , and s 0
that
F;.
~1 +
+
F;.l
t;,
= F;.
n
Since
... +
n
d
B ) =
n
was arbitrary, this proves
is infinitely divisible.
Conversely, suppose that
fixed
d
is symmetrically
n
+ •••
E
F;.
is infinitely divisible and choose for
N some independent random measures
+ t;,
d
n
= F;.
.
is easily seen that
F;.l
d
= ... d= F;.n
such that
By considering the corresponding L-transforms, it
~l'
... ,F;.n
are symmetrically distributed, say with
-128canonical random elements
(aI' 81 ), ... ,(an' Sn) .
~l + . "+~n
from Theorem 9.4 that
+
d
=~
Again we may conclude
has the canonical random element
so we must have
S )
n
d
Furthermore, the random elements
+ S ) = (a, S)
n
(aI' 61 ), ... ,(an' Sn) , being measurable functions of the independent
random measures
Since
n
sl' ... ,sn
respectively, are themselves independent.
was arbitrary, this proves that
In the case
wS
=
00
,
(a,S)
is infinitely divisible.
let us first show that, if
sl
and
s2
are
independent and symmetrically distributed with canonical random elements
(a 2 ' A2)
(aI' AI)
and
elements
(a l + a 2 , Al + A2 )
(a 2 ' A2 )
are measurable functions of
hence that
elements
1;1
and
(aI' AI)
it follows that
1;2
and
respectively, then
has canonical random
sl + s2
To see this, note that
1;1
and
~2
(aI' AI)
and
respectively, and
are conditionally independent with canonical
(a 2 ' A2 ) , given
sl + s2
(a l + a , Al + A ) ,given
2
2
(aI' a 2 , AI' A2)
Using (9),
has conditionally the canonical elements
(aI' a , AI' A2 ) , and the assertion follows
2
by taking conditional expectations with respect to
(a l + a 2 , Al + A )
2
Once this fact is established, we may proceed exactly as in the case
wS
<
The proof is thus complete.
00
According to the last theorem, a symmetrically distributed random
measure
t;
is infinitely divisible iff this is true for the corresponding
pair of canonical random elements.
of
~
In view of Theorem 6.5., the distribution
may thus be described in two different ways by means of spectral
representations of the form (6.6), and we may ask for the relationship
-129between the corresponding canonical measures.
from Theorem 6.5 that, if
(a,A)
~,
or
More precisely, it follows
(a,S)
is infinitely divisible,
then
f
(11 )
t
E
R+
f
E
E
F ($) ,
F (R') ,
+
or
(13)
respectively, where in (11)
M\{O} , in (12)
a
R+
E
o
rEM while
A E
O
M(R:)
A is a suitable measure on
while
M(R:)\{O} , and in (13)
y
is a measure on
is a measure on
N (R') \ {O}
+
and we may ask for the relationship between these quantities.
description of
and
P
X,lJ
QX,lJ
measures on
wS =
by
S
00
R
x
E
and
and
r
S
THEOREM 9.6.
(a o ' AO' y)
or
(aD, y) , let
be the distributions of symmetrically distributed random
with non-random canonical elements
respectively.
wS = 1
the measure on
S , where
A in terms of
For a simple
x
Let
>
M\{O}
(x,lJ)
in the cases
In the former case we further denote
induced by
w via the mapping
s
-+
0
is arbitrary but fixed.
~
be symmetricaZly distributed with respect to
uJiLh canonicaZ random elements
(a,A)
or
(a,S)
.
xc s ,
W E
Further suppose that
Md
-130~
(r,A)
&s infiniteZy divisibZe, and Zet the quantities
(a ' A ' y)
O
o
O
we have in the case
r
(14)
respectiveZy be defined by (11) - (13).
(a ' y)
or
=
wS
wS
r
(IS)
Then
00
A=
=
whiZe in the case
and
fP
x,~
y(dxd~)
+
f Rx AO(dx)
,
= 1
=
A=
J
Qx, ~ y(dxd~).
By comparison with Theorem 9.4, it is seen that the integrals
J Px,~dY
J Qx,~dY
and
are formed in exactly the same way as the proba-
bility distributions of general symmetrically distributed random measures,
except that the "mixing" measures
can in fact be infinite.
y
need no longer be normalized and
In addition we have in (14) a term
J
RxdA O
which can not occur in the corresponding representation of distributions,
since the measure
PROOF.
for any
R
x
is infinite and can not be normalized.
Let us first consider the case
f
log
E
wS
=
00
By Theorem 9.4 we get
F
E[e-~f I
a,A]
= awf
+
J (1
- e-xf(s))(w x A) (dsdx) .
Wri ting
g(x)
we thus obtain
=
J (1 - e-xf(s))w(ds)
x c R
+
-131log Ee
-~f
- - log Ee
.-
-awf-Ag
and hence by (12)
Now
-xc
e
giving rise to the second A-term in (14).
f
5
)w(ds)
Further observe that, by (9),
and so
f
= dy(x,~)
=
f (1
- e-
giving rise to the first A-term in (14).
of the form
rf
with
Next suppose that
f
E
F
r =
wS
f (1
mf
)
- e-mf)p
x,~
(dm)
=
J p (dm)dy(x,~),
x,~
Finally, the term
aOwf
is
This completes the proof of (14).
=
1
=
We then get by Theorem 9.4 for any
-132Ee-
sf
=E
rj 8.f(T.))
J
J
exp(-awf -
= Ee -aw
f
II
E(e
-8.f(.. )
J
JIB.]
j
=
Ee
-awf
exp
= EE (exp (-awf
LB. f (T .) I
j
J
J
a, 8]
=
J
L log
E[e
-8.£(T.)
J
J
j
=
-
Ee -awf exp (-\ ~' g(B j ) )
=
I
8.]
=
J
Ee -awf-8g ,
J
where
g(x)
=-
log
J e-xf(s)w(ds)
,
R+
X E
Using (13), we thus obtain
But from the above calculation it is seen that
II E N(R~)
,
and hence
= J dy(X,ll)
= J (1
- e-
J (1
mf
)
Hence (15) holds, and the proof is complete.
- e-mf)Q
JQ
X,ll
X,ll
(dm)
=
(dm)dy(x,ll).
=
-133-
r
We are now able to characterize the class of measures
and
A
."
which can occur in the representation (11) of infinitely divisible symmetrically distributed random measure.
THEOREM 9.7.
Let
Write for brevity
K'
Md and rEM, and let A be a measure on M\{O}
W E
Then there exists some infinitely divisible random measure
symmetrically distributed with respect to
in the case
(i)
1..
E
0
AK
(16)
0
(ii)
M(R')
+
<
on
y
Y{).IK'
Suppose that
00
,
wS = 1
= oo}
The quantities
PROOF.
=
(14) holds for some
I
~
(1
(R~))\{O}
(a ' 1.. ' y)
O
0
~
a , some
a
O
~
satisfying
<
00
0 and some
satisfying
or
(a ' y)
O
are then unique.
is an infinitely divisible symmetrically distributed
Then (11) must be the canonical representation
occurring in Theorem 6.5, and therefore
r
and
A are necessarily unique.
But then (14) or (15) holds by Theorem 9.6 for some
satisfying (12) or (13) respectively.
get in particular
O
= 0
random measure satisfying (11).
(u ' y)
O
a
_ e- X-).IK)y(dxd).l)
(15) holds for some
(R+ x
which is
(R+ x M(R~)J\{O}
Y on
Y{).1 K = oo} = a
&n the case
(17)
wS
~
wand satisfies (11) iff
and some measure
00
measure
(x) :: x .
(a ' 1.. 0 ' y)
O
In the case
wS
or
=
00
we
r
- 1og Ee -t(a+AK)
(18)
and since
AK
<
=
a0t +
from (18) by monotone convergence as
completes the proof of (16).
t
t
+
In the case
Furthermore, it follows
0+
wS
that
=1
a.s., and use a similar argument to prove (17).
o'
or
A ' y)
O
0 ,
a.s. by Theorem 9.4, the last two terms on the right-
00
hand side of (18) must be finite for all
(a
~
t
(a ' y)
O
0
= Y{~K = oo}
, note that
SKI
,
which
<
00
The uniqueness of
follows by the uniqueness assertions in Theorems
6.5 and 9.4.
Conversely, let
r
A by (14).
and
in (16).
Note that
A exists according to the second relation
By Theorem 6.5 and the last relation in (16), there exist some
random variable
(a,A)
be such as described in (i), and define
(a ' A ' y)
O
O
a
~
and some random measure
0
on
is infinitely divisible and satisfies (12).
vergence as above, it follows from (12) that
by Theorem 9.4,
(a,A)
has to be infinitely divisible.
~
<
such that
Using monotone con00
a.s., and hence
can serve as the canonical random elements of
some sYmmetrically distributed random measure
related to
AK
R'+
~
which by Theorem 9.5
Hence by Theorem 9.6,
r
and
A are
by (11), and therefore have the asserted property.
proof works in the case
wS
=1
A similar
.
NOTES. Theorem 9.1 is known for t
=
00
in which case it was proved,
independently, by Davidson (1974) (in a special case) and Kallenberg (1973a).
The latter paper is also the source for Corollary 9.2, which extends a result
by Nawrotzki (1962)?
Furthermore, Corollary 9.3 is a typical resul t from
Kallenberg (1974a), while Theorem 9.4 was proved by BUhlmann (1960) and
Cf. Kerstan, Matthes and Mecke (1974), p. 80.
-135-
by Kallenberg (1973b) and (1974 b), (see also Kallenberg (1973c)).
Finally,
Theorem 9.5 extends two results for point processes given by Kerstan,
Matthes and Mecke (1974),
new.
pages 66-67, while Theorems 9.6 and 9.7 are
7
PROBLEMS.
9.1.
Let
S
{I, ... ,n} , let
=
w be counting measure on
~
be counting measure on a subset of
at
rando~.
Show that
distributed w.r.t.
k
=
0 or
n).
k
elements in
S and let
S chosen
is (in the obvious sense) symmetrically
~
wand still not a mixed sample process (unless
Thus the assumption
w
E
Md is essential in Theorem
9.1.
9.2.
Let
~
be a sample process on
Lebesgue measure.
[0,1]
with intensity n
times
Show that there does not exist any symmetrically
n on some
1 , such that the restriction of n to
distributed (w.r.t. Lebesgue measure) point process
interval
[0,1]
n
>
=0
~.
(Hint:
n[O,l]
cannot be non-random
a.s.)
Extend Corollary 9.2 to general symmetrically distributed random
measures.
9.4.
c
is distributed like
unless
9.3.
[O,c],
Let
w
E
S ~ O.
(Cf. Westcott (1973).)
Md and suppose that
is a S-compound of n , where
Show that ~ and n are simultaneously symmetrically
distributed w.r.t.
w
Prove the corresponding result for the
random measures defined by (8.11) and (8.12).
(Hint: use analytic
continuation. )
9.5.
7
Md be fixed and let
Let
W E
W •
Show that
p~-l
~
be symmetrically distributed w.r.t.
is uniquely determined by p(C~)-l for any
Extensions of Theorems 9.5 - 9.6 are proved in Kallenberg: "Infinitely
divisible processes with interchangeable increments and random measures
under convolution", Tech. flep0Y't, Dept. of Mathematics, Gljteborg Universlty.
-136C € B with
fixed
wC
>
O.
(Hint: use the connection with thinnings
exploited in the proof of Theorem 9.4.
9.6.
10.
Specialize Theorems 9.5 - 9.7 to the case of point processes.
PALM DISTRIBUTIONS.
Let
be a random measure on
t;
(p ~~-l)l
Camp b e II measure
S , and define the corresponding
S x M by
on
M € M.
B€ B
(1)
If
Cf. Ka11enberg (1973c).)
Et;
is a-finite, which holds e.g. when
corresponding Radon-Nikodym derivatives
X
P M = (pt;-l)l(ds M)
s
(pt;-l) 1 (dsxM)
= ~[t;(ds);t;€M]
P M by
s
S
Et;(ds)
Et; € M , we can define the
€
S a.e.
f € F is such that
More generally, suppose that
Eft;
M € M.
is a-finite.
We can then define
; t;€M]
Ps M = E[ft;(ds)
Eft; (ds)
S €
S
a.e.
Eft;
without ambiguity, since the two expressions for
a.e.
Et;
on the support of
of M w.r.t.
s
M E M and that
for fixed
s
holds a.s.
Et;
S
.
We shall call
Note that, by definition,
for fixed
E
f
Ps M clearly coincide
Ps M the Palm probability
P M is 75-measurab1e in
s
P M is almost a probability measure in
s
M
, in the sense that every probability defining property
Just as in the case of conditional probabilities, we
s
-137-
M (to
may choose a version which is actually a probability measure in
be called the Palm distribution) for every fixed
s
E
This is possible
S.
by the usual arguments for construction of conditional distributions (cf.
Bauer (1972)), on account of the following fact.
LEMMA 10.1. M and N are Polish spaces in their vague topologies.
PROOF.
By Lemma 4.4, it is enough to consider
base in
S , and assume without loss that
Approximate each indicator
~
f.
n J
J
compactness of
Nil
j
such that
E
~n
~f.
{~n}
v
, so any sequence
~'
(n
~ E
Furthermore, we have
E
Nil)
N'
c
N must contain a subsequence
f. -+ ~ If. (n E Nil)
n J
J
, proving that indeed
But then
N , so we get ~f. - ~ 'f. and hence ~ = ~ ,
J
J
v
It is now easily seen that the function
-+ ~
~
2
p: M
-+
R
+
defined by
00
p(~, ~')
= L
~, ~'
EM,
j =1
is a complete metric on
M is separable, let
R:
a
respectively.
E
A,
b
E
M generating the vague topology.
A and
M
In fact, the latter condition implies
j EN.
some
~n -+
j EN.
J
for all
-+ ~f.
G , from below by a sequence
E
G E G , and note that, by Dynkin's theorem, each
is uniquely determined by
iff
G
be an enumeration of all these
c
functions for all
G be a countable
Let
G is a semi-ring.
lG'
F
of functions in
M.
To see that
B be countable dense subsets of
Then the class of all finite sums of measures
B , is clearly countable and dense in
M.
Sand
bo a ,
,
-138-
Henceforth, we shall always assume the set functions
bilities on
M (and even on
~
N if
p
s
to be proba-
is a point process), and for con-
venience of writing, we introduce for every
~s
p
' defined on the same probability space as
s
E
S
~
a random measure
and with distribution
s
Just as in the case of conditional distributions, the expectations
corresponding to Palm distributions can be computed in two different ways.
More precisely, we have the following result, known as Campbell's theorem.
LEMMA 10.2. Let
let
f: S
x
~
be a random measure on
M-+ R be measurable.
+
Ef(s, ~s) = f f(s,
Q
We have to prove that
JB Ef(s,
S E
Ef(s, ~s)
~ )E~(ds) = E
s
Now this is true by definition of
A
x
B,
A E B,
a-finite, and
~ s (w))P(dw) = JM f(s,~)P s (d~) =
EHds)
(2)
E~
Then
= Ef(s,~)S(ds)
PROOF.
S with
J
B
~s
S a.e.
E~.
is B-measurable and satisfies
f(s,~)S(ds)
for indicators
B
E
f
B •
of product sets
M EM, and it easily extends by Dynkin's theorem to
indicators of arbitrary sets in
B x M.
But then it holds for simple
functions, and finally, by monotone convergence, for arbitrary
f .
-139-
PF;-l
In the particular case of simple point processes,
interpreted as the conditional distribution of
should be
s
F;, given that
F;{s} =
1
(Note, however, that a definition along these lines only makes sense when
p{~{s}
o}
>
We could then expect
> 0.)
F;s
to have an atom at
s , and
this can in fact be achieved by changing the definition on an EF;-null
set:
LEMMA 10.3.
Let
F;
there exist versions of
each
S
S •
E
S with EF; a-finite.
be a point process on
such that
F;s
+
S
f(s,~)
PROOF.
(4)
we then have
f: S x N + R
(3)
where
is a point process for
F;s - Os
For any measurabZe function
may be arbitrariZy defined for
E
S a.e.
~
tN.
E~,
Clearly
{(s,~) E
S x M:
~
-
°s
E
= (S
N}
x
N)
n {(s,~)
E
S x N:
B c B be arbitrary and let
B
{ (5,11)
{B .}
nJ
~{s} ~
B
To see that the second set on the right of (4) belongs to
of
Then
x
I} .
N , let
B be a null-array of partitions
c
Then
E
B x N: jJ{ s} ~ l}
=n
u (B . x {jJ EN: jJB . ~ I}) to B x N ,
n j
nJ
nJ
and the measurability in (4) follows from the fact that
By (2) and (4) we now obtain for any
B
E
B
S
is a-compact.
-140-
fB P{~ s
- 0s E
N}E~(ds) =
E
IB l{~_o s EN}(s,~)~(ds)
=
=
E
IB 1{ i;; {s hll (s ,0 i;; Cds)
=
E!;B ,
so
P{~ s
fB (1 -
- 0
N})E~(ds)
E
s
=
0 ,
and hence
p{!;
(5)
Redefining
hold for all
i;;s
==
s
- 0
s
sand
For each
s E S
and
B E
w.
a.e.
E~.
on the exceptional s-set in (5), we can make (5)
0s
the exceptional w-set where
all
=1
E N}
Since
~
S
s E S
i;;s - 0 s
we then redefine
4N
to make
~s -
~s =
<5
s
<5
s
E N hold for
B - 8 B remains A-measurable for each
S
B , it is now a point process.
on
s E S
But then (3) is an immediate
consequence of Lemma 10.2.
Besides the random measures
f
E
F with
E~f
E R'
+ '
i;;s
we shall also consider, for arbitrary
~f
the mixture
with distribution defined by
M EM.
A
When
~
is a point process, we also consider the point processes
with distributions defined by
P{~f E M} =
E~f Is
P{!;s
=
E~fEIs
l{i;;_o EM}(s,i;;)f(s)~(ds)
8
5
5
E M}f(s)Eqds)
MEN .
f,f
-141-
~f
It is easily verified that Campbell's theorem remains valid for
and
A
~f
' in the sense that
Eh(sHf
(6)
E~f
Here
h
is any measurable function from
N respectively to
M or
R
+
From the definitions of Palm distributions of a given random measure
it is obvious that they determine, together with the intensity
~
the Campbell measure
(Ps-l)l
E~,
p~-l .
and hence also the distribution
Indeed, a much stronger result is true, indicating a close relationship
between the Palm distributions for different
LEMMA 10.4.
Let
R: .
Then
E~f E
to the set
a.e.
E
Esf and
{~: ~f >
O}
pE,;-l
f
E
S:
and let
S
f
E
F with
determine the restriction of
p~-l
They further determine
s
p~-l
for all
s
E
S
E(fE,;)
Note that
S
be a random measure on
~
s
S
Then
f
may always be chosen such that
~f =
0
implies
~
=
a ,
f(s) > 0
so in this case
alone determine the whole distribution of
~
Esf
for all
and
.
PROOF. The first assertion follows from (6) by choosing for fixed M E M
~f >
~f
0 ,
=0
,
-142-
since for this
h, (6) takes the form
p{i; E M, i;f
(8)
= Ei;f Eh(l;f) .
> O}
Furthermore, the probabilities in (8) determine the measure
E[ (f S) B; I;
E
E[Cfi;)13; i; E M, H
M]
13
0]
>
B
E
and so the second assertion follows from the fact that
P{I;
s
E[i;(ds) ;I;EM]
EI;(ds)
M}
E
For
f E F
c
E((fS) Cds) ;i;EM]
E (fl;) (ds)
S
ES
a.e.
E(fS) .
, the above uniqueness assertion can essentially be
strengthened to continuity:
THEOREM 10.5.
f ( F
c
be such that
condition
n
+
PROOF.
Ei;f E
be random measures on
Under the assumption
R~
then implies
d
I;nf
Furthermore ~
implies
EI; f
1;, 1;1' 1;2'
Let
+
I;
n
~
I;f
!!
EI; f
n
whi le conversely
I;
and
Ei; n f
+
I;nf
d
+
i;f
and let
S
+
i;nf
the
~
El;f
d
+
i;f
imply that
El;f .
First assume that
be hounded and continuous.
i; n ~ I;
and
Then the mapping
Ei;f , and let
p
-*
h(p)pf
h: M+ R+
is continuous,
so we get
h(1; )F, f ~ hCF,)r,f
(9)
n
n
by Theorem 5.1 in Billingsley (1968).
relations
F, f
n
~ i;f and Ei; n f
+
Er,f
Furthermore, it follows from the
that
{E; f}
n
is uniformly integrable
-143-
(cf. Theorem 5.4 in Billingsley (1968)), and hence so is
{h(~
n
H n f}
.
By (9) we thus obtain
(10)
n nf
Eh(~ )~
+
,
Eh(~)~f
so by (6)
(11)
Since
h
d
was arbitrary, this proves that
d
Suppose conversely that
~nf + ~f
is bounded and measurable with
~f ~
Of
~nf +
~f
E~nf + E~f
and
If
h:M+R
+
a.s., it follows by Theorem 5.2
in Billingsley (1968) that (11) holds, and by (6) this implies (10).
g: S
For any fixed bounded and continuous function
h (]J)
In fact,
h
__ {]Jo (fg) /]Jf
]Jf
>
+
R
+
we may choose
a
is bounded and measurable, and by (6) the set
0h
{]Jf
=
=
O}
satisfies
For this particular
E(f~
and since
g
as asserted.
h , (10) takes the form
n
)g =
E~
n
(fg)
+
E~(fg)
= E(f~)g ,
was arbitrary, it follows by Theorem 4.10 that
f E:
wu
n
--)-
fE: ,
-144-
Finally suppose that
h(p) ::: (1
).If) -1 .
+
Then
~n
i ~
h().I)
d
and
and
~nf + ~f
h().I)).If
tinuou5, so (10) and (11) are satisfied.
it follows from (6) that
E~
n
f
-+ E~f
' and choose
are both bounded and con-
Since moreover
0 ,
Eh(~f) >
.
In case of point processes, we may also prove a continuity theorem
A
for
~f'
f
E
THEOREM 10.6.
E~
that
E
Fc
Let
be point processes on
S
~
and suppose
Then any two of the following statements imply the
M\{O}.
third one.
r X. Er<"
E<'n
(i)
(li)
(i ii)
PROOF.
Let
d
A
~nf
f
-+
E
A
~f
for all
f
Fc and let h: N +
E~f >
with
F
c
E
0 .
be bounded and continuous.
R+
Then
the function
(12)
g().I) =
J h().I S
)l n
'!
).I
In fact, suppose that
is vaguely continuous.
n EN, and that
0 )f(s)).I(ds)
s
11
,..
in
).I
Nand
n
- 0
s
s
n
-+
5
eN,
).I{s}? 1
in
S
and
).I {s }
n
Then
n
holds hy definition of vague convergence, and hence hy continuity
n
> 1 ,
-145-
h (]J
Since
hand
(1968) that
f
g(]Jn)
only assume that
n
- 0
s
) f (s )
n
n
-+
h (]J - 0 ) f (s ) .
s
has bounded support, it follows by Theorem 5.5 in Billingsley
-+
g(]J).
h (x )
n n
-+
(Note that, in Billingsley's notation, we need
hex)
for
x
-+
n
=
x
such that
Eg(t;)
n
1 Eg(t.:)
o.
Since
h
h
and
I:h(t;nf)
-+
x
n
provided
g
by (12) with
EN.
is bounded and continuous, so by (iii)
A
(13)
Moreover,
Eh(snf)
g
-+
Eh(€f) > 0 .
is continuous, and it is even bounded since
f(s)Jl(ds)
l+Jlf-f(s)+llfll
f(s)}l(ds)
l+Jlf
]Jff
l +]J
<;
1 .
IIence by (ii)
(14)
E
En
be as above.
was arbitrary, this implies (iii).
]J
h
g
Eh(t;f)
Next suppose that (ii) and (iii) hold, and define
Then
E and
follows by uniform integrability as in the preceeding
proof, and it follows by (i) and (7) that
Et.;f>
E
1 .)
Now suppose that (i) and (ii) hold, and let
Then
x
Eg (s )
n
-~
Using (7), (13) and (14), we obtain
Eg (S) .
Et:nf
-+
Et;f.
If
U:f
0, we
m~IY
-146-
choose
fl , f2
Fc
E
with
f
=
f
1
- f2
E~fl = E~f2 >
and
0 , and conclude
that
.
E~f
E~
Thus
n
f
+
E~f
is valid for all
f
E
F
c
,and (i) follows.
Finally suppose that (i) and (iii) hold.
f
Fc
E
fe
E
-tf
I
S
h: N + R+
and any bounded continuous
For any fixed
f
and put
exp{-
t~
n
E
F
h(ll)
f
+
and
c
==
e
t
-tllf
in (15) by
f
, thus obtaining
tf(s)}f(s)e-tf(s)~
+
we may replace the
R+ '
E
By (7) we get for any
E
J
n
(ds)
t~f
exp{-
s
+
tf(s)}f(s)e-tf(s)~(ds)
,
or equivalently
E~ fe
-t~
n
f
n
E~fe-t~f .
+
By Fubini's theorem and dominated convergence, we get
He
-~ f
n
=
1 _ E
II
a
~ fe
n
-t~ f
n dt
=
1 -
l
Jo
= 1 - I'
and since
f
E( fe
n
I;,
-t~ f
n dt
+
1 -
II
EE,fc -tf;fdt
a
UC -tcf Jt " Eo -(I' ,
was arbitrary, it follows by Theorcm 4.5 that
r >-d r,
'n
-147This completes the proof.
Palm distributions are often easily calculated by means of L-transforms.
In fact, we get by dominated convergence for any
f
El;f
with
F
E
E
R'
+
(16)
and
Ll;
is obtained from this derivative by normalization.
As a: simple
s
example, suppose that
is a Poisson process with intensity
l;
A
E
M\{O} .
By dominated convergence we get
so putting
t = 0 ,
and hence by normalization
L
r (f)
J
=
Ll;(f)e-f(S) = Ll;(f)L
s
El; , which yields
E;f
d
(f) = Ll;+8 (f)
E;
S
El;
S a . e.
E
s
d
E;s = E; + 8 s
By Theorem 3.1 we thus obtain
a.c.
85
for every
or equivalently
f
E
F with
Et;f
d
t: s - 8 s
E"
R'+
(
We
shall usc this fact to prove the following complement to Corollary 7.4.
A ( r~\ {O} , Ze t;
I ( B
of point processes on
A
s
be a DC-serni-ring and Zrt
Put
sn = L. l; nJ. •
J
Then
{F,
t;
l'e a null-or-rlay
.}
nJ
n
~
t;
and
-148-
for aU
f
F with
E
L P{I; nJor
ei)
El;f
>
O}
R'+
E
iff
r
-+ AI
I ,
E
j
L E[E; nJ B;
(i i)
0
j
PROOF.
F,njB
> 1]
-+
B ( B
0
Suppose that (i) and (ii) hold.
E,
Then
d
n
E,
-~
follows by Corollury
7.4 since
L P{E,
(17)
j
08 > 1}
nJ
~
}
~
E[I; .8; I; .8
nJ
nJ
J
>
Furthermore, we get by ei), eii) and (17) for any
L
EI; .r
nJ
j
:=
L P{I; nJor>
j
O} -
L
P{I; or>
j
nJ
l}
L EE,
XA
+
B
1] -+ 0
I
E
L
E[I; 01;
j
nJ
E,
E
B .
I
or>
nJ
1] -+ AI ,
proving that
:=
j
0
nJ
lienee it fa llows by Theorem 10.6 that
EE;f
E
A
E;nf
EE; •
d
A
-+
t.f
d
for any
E;
f
(
F with
R'+
Suppose conversely that
f (F
with
~
<,
J
EE;f
P{t; .8
nJ
>
E
OJ
d
-+
I;
and that
d
A
I;nf
-+
S
d
A
sf
for any
By Corollary 7.4 and Theorem 10.6, we get for
R'
+
-+
I;n
AB
L P{E; nJ.B
> I} -+ 0
o
J
Ilere the fi rst relation contains (i)
L U: nj 8,.
j
and moreover
AB .
-149-
L. Efi; nJ.B;
~ .B > lJ
nJ
]
= L E~
j
.B -
nJ
L
P{~ .B
j
nJ
O}
>
Ij
+
P{~ .B > I} ~ 0 ,
nJ
proving (ii).
NOTES,
on
R
Palm probabilities were introduced for stationary point processes
hy Palm himself (1943), whose studies were continued by several
v
authors, including llinc'in (1960) and Slivnyak (1962).
The present approach
is essentially due to nyll-Nardzewski (1961), (see also .Jagcrs (197:\),
Papangelou (1974), and Kerstan, Matthes and
~1ccke
working directly with the Campbell measures).
(1974), the latter ;Iuthors
Camphell 's theorem, in the
formulation of Lemma 10.2, was first stated by Kummer and
while Lemma 10.3 is implicit in Kallenberg (1973a).
10,.4 is new.
~Iatthes
(1970),
The uniqueness Lemma
As for the corresponding continuity problem, a discussion
for stationary point processes is essentially given by Kerstan and Matthes
(1965), and for general point processes by Kallenberg (1973a), from which
Theorem 10.6 and Corollary 10.7 are taken.
Theorem 10.5 extenns a result
in the same paper.
PROBLEMS.
10. 1.
~illjngsley
Prove the version of Theorem 5.5 in
(1068) which
1S
used in the proof of Theorem 10.6.
10.2.
M
E
M
Let
Show that
x -1
E
M}
= 1
be a random measure on
E;
and let
=
P{i;
S
with
Ei;
P{i;
iff
s
a-finite, and let
E
M}
=1 ,
E
S
U,B <
m
S
Ei;
a.e.
10.3.
be a random measure on
~
Let
x
>
0
be fixed.
JB p{i; s B = x}Ei;(ds)
S , let
Show that
B c B
P{i;B
x}
, (cf . .Jagers (1973)).
with
0'
-150-
10.4.
Let
F;,
and
En
Pn
10.5.
n be independent random measures on
and
M
in
Calculate
P (F;,
-1
n) s
+
-1
with
pF;,-l
s
S
in terms of
T:r,
and
s
Let
on
F,
M
on
by
(T
s
~)B = ~(B -
s) ,
B
B c
stationa~y if P(TsF;,)-l
R is
Show that in this case
EF;,
s
,
S
E
S ,
Say that a random measure
1S
independent of
s E R .
A times Lebesgue
equals some constant
-1
A E R' , then
measure, and that if
a.e.
T
and define the translation operators
S = R
peT F;,)
is independent of s
s
The converse is also true. (Cf. Mecke (1967) and Jagers
EF;,
+
(1973). )
10.6.
Let
be a random measure on
~
R
and let
f
E
F (R)
Show that it is possible to define a random measure
with
~f
on
R such
that
f P{T -s F;,
J
5
E M}f(s)EF;,(ds)/EF;,f
=
E
f
l{T
-5
~1 EM.
F;, EM}fF;,(ds)/EF;,f
s
State and prove the corresponding version of Campbell's theorem.
10.7.
State and prove a version of Theorem 10.6 for the random measures
F;,f
defined in Problem 10.6.
random measures.
Specialize to the case of stationary
(Cf. Kerstan and Matthes (1965) as well as Kallenberg
(l973a) . )
10.8.
Let
F;,
be a random measure on
EE,f c R'
+
on
R
+
x
S
and let
f (F
be such that
Define the di stri hut i on of a random element
M by
(n ,
f
r: f )
-151for any measurable set
A in
R
+
M , and show that this probability
x
equals
State and prove the corresponding version of Campbell's theorem.
11.
CHARACTERIZATIONS INVOLVING PALM DISTRIBUTIONS.
Tn the preceding section we saw that any Poisson process
S
~s -
satisfies the relation
0
s
~ ~,
S
E
E~.
a.e.
S
[,
on
Actually,
this relation characterizes the class of Poisson processes among arbitrary
random measures:
Proposition 11.1.
~
Then
PROOF.
n (
~
Let
be a random measure on
&s a Poisson process iff
Suppose that
~
d
s
~
0
+
s
~
d
s = E.' + 8 s
a.e.
with
S
S
E
S
EM.
E[,
a.e.
Then we get for any
Et; .
B
E
Band
N
n} =
!-E{~B;
[,B
n
=!-J
n
13
P{[,B =
n} =
n -
!n
JB E[[,(ds);
l}E~(ds)
t;B
= n] =
!- P{~B
!-
f
P{[, B
nBs
n}E[,(ds)
n - l}E~B ,
n
and hence by induction
P{~B = n} = P{~B
O}
([[,13/
n!
In the same way we get more generally for any
kEN, nl' .. , ,n k
E
Z+
..
-152and disjoint
B , ... ,B
k
l
E
B
n.
k
P
(EI;B. ) J
k
k
J
n {I;B. = n. } = PH u B. = a} IT
J
J
j =1
j =1 J
j=l
P{~B
/\ similar argument also shows that
(n.) !
J
z+ } = a ,
1
B
B .
E
Hence
must be a Poisson process, and the proof is complete.
The aim of the present section is to prove extensions in various
directions of the ahove proposition.
We shall first consider the case
of infinitely divisihle random measures
with
i;
EM.
Ei;
If
a
and
A are the corresponding canonical measures defined in Theorem 6.4, we
can define the corresponding Campbell measure
A (B x M) = aBeaM +
A = ae
Formally, we may write
Since
J lJBA (d~)
JM
= EI;B <
a
~BA (d~)
+
B
AM
s
f
A(dsxM) =
A(dsxM)
on
S x M by the relation
ME M
B
J
A
s
s
E
S
a.e.
by
a(ds)+fM~(dS)A(dlJ)
A
S
i.e. we have for any non-
S
a(ds)f(s,O)+ f(s,lJ)lJ(ds)A(dp)
a.e.
a(ds)+
EI;
a(dS)OoM+JM~(dS)A(d~)
Af
s
pA (dlJ) , d. (10.1) .
we can proceed as in the definition of Palm
00
Campbell's theorem remains valid for the
negative function
E
Al =
Al , where
distrihutions and define the distributions
=
on
A
lJ(dS)A(d~)
U:.
-153-
Let us now use the method of the preceding section to calculate
For any
f
F
E:
t
E
R+
and
a~
= L~(f
by dominated convergence.
B
E
s
B we obtain
exp{- af - taB -
+
p
tlB){aB
+
f
(1 -
e-~f-t~B)A(d~)}
J ~Be-~f-t~BA(d~)}
Hence
and so, by Campbell's theorem,
L~ (f) = L~ (f)
s
Since the last expression is the L-transform of
p[-l = p~-l*i\
's
s
by Corollary 3.2 that
p~-l
I
p~-l
s
( p~-l
1S
it follows
a. e. , and in particular
pt;-l )
s
a convolution factor of
.
This factorization
property characterizes the class of infinitely divisible random measures:
THEOREM 11 .2.
Let
~
be a nxndom measure on
&s infinitely divisible iff
rr- l
c(we
'5
1\
case:
S
z.,)i th
s ( S
H: (M
Then
a.e.
a.c.
much stronger result, to he proved first, holds in the point process
-154-
THEOREM 11 .3.
E;.
Let
be a point procesB on
E;.
is infinitely divisible iff P{E;.f
for every
f
PROOF.
If
P(E;.f) -1
I
E;.
is infinitely divisible and
P(E;.ff)-l
6.1 yields
P{E;.B
=
O}
B
> 0
f
E
F
c
~'(t)
EF;f
for some random variable
to
f
E
F
c
B , which implies
~ =
'l'hen
Furthermore, Lemma
P{E;.f = O}
E;.f ~ 0
>
0 ,
f E
F
c
and suppose that
t
2>
0 ,
~(t)
Z+ ' so if we divide by
E
E;.f ~ 0 , then
with
LE;.f ' it follows easily that
~(t) Ee- nt
=
n
E
be fixed with
Writing
0
M
E
P(E;.f)-l
follows by the above calculations.
Conversely, let
from
and
> 0
EE;.
E;.f ~ 0
with
F
c
E
O}
=
S l,lith
and integrate
t , we obt'ain
-1T
at
A(l - e
+
t)
where
a
= Pin = O}EF;f;
A(dx)
By Lemma 6.1, this proves that
x
E;.f
-1
Pn
-1
(dx)EF;f
x > 0 .
is infinitely divisible.
The sufficiency
assertion now follows by Problem 6.5.
Proof of Theorem 11.2.
To show that
pr'ove that
~
Suppose that
IH-s
l
I
-1
Pi';,
s
S
f:
S
a .c.
,: F,
•
is infinitely divisible, it suffices hy Lemma 6.4 to
U:;B , ... ,E,B )
I
k
and every disjoint
IS
infinitely divisihle for every
'\, ... ,1\ ( 13 .
Since clearly
Jl r.
-1
k ( N
for
-155any
B
B with ~B ~ 0 , it is enough to prove the theorem for finite
E
S , say for
S
let
n'
and
n
=
{l, ... ,k}.
and hence also
c
Pn-
1
respectively.
I
PCn'
and
0
>
s
E
c C~{l},
be Cox processes directed by
'~s{k})
cCE;s{l}, ...
For fixed
, ~{k})
From Cl.4) it is seen that
Writing
+ 0 )-1
s
~
=
Lc~
~{s} ~ 0 ,
S with
and
Pn-
l
I
Pn,-l
, we further obtain
from C1. 4)
~ ~ (l-e
-t
-t
,l-e
1,
~~
CO,
k)
-t
e
,0)
s
t 1,
n'
and so
+ 0
arbitrary with
every
f
Since
~{s} ~ 0 , we may conclude that PCnf)-l
F , and it follows by Theorem 11.3 that
c
E
divisible.
8.6 that
proving that
S
The factor
t;
c
>
0
,t
n
S
I
k
E
R+
S was
E
PCnff)-l
for
is infinitely
being arbitrary, this implies by Lemma
itself is infinitely divisible, and the proof is complete.
We are now going to show that it is enough in Theorem 11.2 to consider
one particular mixture of Palm distributions, provided we add an assumption
about the constant part of
~.
Given any
f~
define the non-random set function
= inf{x
>
0:
P{(f~)B
< x} >
o}.
superadditive and continuous at
to ensure
n
to be a measure.
Since
~,
on
ft;
f
E
F
with
t;f
<
00
B by ft;B = essinf
a. s., we
(f~)B
=
is automatically finitely
only finite subadditivity is needed
This holds in particular if
essinf E;f = 0 .
-156-
THEOREM 11.4.
Let
positive with
0 <
~
be a random measure on
E~f <
S and let
F be strictly
f E
is infinitely divisible iff
00
f~
is
finitely subadditive and
PROOF.
~ ~ I(a,A) . Then f~ = fa EM, (this is a simple
Suppose that
special case of Theorem 6.9), and it follows in particular that
f~
1S
subadditive.
Suppose conversely that
p~-l
and that
I p~il.
Since
it is easily seen that
and
~
is subadditive (and hence a measure)
f~
f
is strictly positive by assumption,
are simultaneously infinitely divisible,
f~
(~Bl""
to prove that
,Bk
B.
E
,~Bk)
= IS'
f = I
and so it is enough to assume that
By Lemma 6.4, it suffices
is infinitely divisible for every
kEN
From the assumed factorization property it is
seen that
E[~S exp(-L.t.~B.)]
E~~ J J
= E exp(- r.t.~B.) E exp(~ L.t.n·)
•
JJ
J
JJJ
~
for some R+-valued random variables
~
n l , ... ,nk ' and writing
~
- I
we get
E[~S exp(-L.t.~B.)J
(1)
J J
J
----~......<---"--~
=
E exp ( - Y. . t. ~~B. )
'J J
E~S
J
E~S E exp(-L.t.n.)-~S
J J J
E~S
Thus the numerator on the right is positive, and letting
we obtain
E~S
P{n
l
= •••
=n
k
= O}
2
IS.
-
t , ... , t
l
k
-~
00
But then the second factor on the
right of (1) must be an L-transform, and (1) shows that the assumed factori~
zation property of
now on that
~
= 0
~
remains true for
~.
We may therefore assume from
-157-
Let us now suppose that
{O,l, ... ,k}
=
11
(11
0
,
11
1
(I;;S, I;;B ,
directed by
erating function of
(2)
n
,11
,I;;B )
I
k
proof of Theorem 11.2, it is seen that
the Palm distribution of
, . o.
Pll-
1
I
w.r.t. the point
k
)
0
is;) Cox procc s s on
."
Proceeding as in the
Pll,-l
where
0, and if
Pn,-l
denotes
is the gen-
~
n , we obtain
t ~ I (t, t 1 ' ... , t k )
t, t
Ei;;S
l
, ... ,t
k
E
[0,1] ,
for some Z+-valued random variables
S, sl' ... ,sk ' where the prime stands
for differentiation with respect to
t.
11
HO,
t
l
,
'0.
,t k ) = E[t
so it follows from (2) that
right, i.e. that
P{s =
a} = a
t
l
l
is a factor of the expectation on the
t
1/J'(t,t l ,···,t k )
H t ,t 1 ' . . . , t k)
and integrating w.r.t.
Now
from
Rewriting (2) in the form
=
1 sl
l
s
EI;;S Et - t
a
to
t
sk
k
1, we thus obtain
In particular
a
- log 1/J(I, ... ,1)
- log HO,I, ... ,1) -
E~S
Es
-1
and hence by subtraction
- log 1/J(1, t ,
l
log HO,I,
,t )
k
=
,1) - log 1/J(0, t l , ... ,t k)
+ I:F;S
l
1
Es - (1 - t:
-158-1
Es
Since
[O,l]k
<
00
defined by
t
1S
on
l
, ... ,t
k
[0,1] ,
E
the generating function of some infinitely divisible random vector
Zk
+
be a Cox process directed by
Letting
~p
and writing
for any
(3)
where
p
- log
=
E
for its generating function, we obtain more generally
(0,1]
~p(l,
tl,
,t k ) =
log
~p(O,l,
,1) - log ~p(O, t
G
p
is an infinitely divisible generating function.
by Corollary 8.5 a p-thinning of
where
q
= 1 - p.
infinitely
(4 )
e on
it is seen from Lemma 6.3 that the function
,
n
p
l , ... ,t k ) - log
~(tl'
But
... ,t k )
n
is
, and so by (1.6) and (3)
Here the last term corresponds by Lemma 8.6 to an
divisible distribution, so if we can show that
log t/Jp(O,I, ... ,I) - log
~p(O,
q
+
pt l . . . . • q
+
pt k )
+
0
as
p
+
0 ,
-159it will follow that
~(l,
t l , ... ,t )
k
is the limit of a sequence of
infinitely divisible generating functions, and hence itself infinitely
divisible.
¢ = L
To prove (4), put
~S, ~Bl ' ... , ~Bk
E
exp{-p-l(~S(l
= cpp
(
-1
(1 - t), P
, and note that by (1.4)
L.~B.(l
J J
- t) +
-1
- t.))}
=
J
(1 - t 1)' ... , p
-1)
(1 - t k )
Hence
~
P
(0,1, ... ,1) = ep(p
-1
, 0, ... ,0) = E exp (-p
-1
•
~S)
while
~ ~ P (0 , q • ...
=
, q) = cp (p
-1
,1, '"
E exp(-p-l~S - L.~B.) ~ E exp(_(p-l + k)~S) .
J
J
Now it may be seen as in the proof of Theorem 11.3 that
and since
~
=
0
we have
a
=
,1 )
O.
d
~S =
some
l(a,>') ,
Thus by Lemma 6.1, the difference in (4)
is at most equal to
log E exp(_p-l~S) - log E exp(_(8P-l
which tends to zero as
that the Cox process
p
+
0
+
k)~S)
=
by dominated convergence.
(n l , ... ,n k )
directed by
This proves
(F,3 , ..•
1
,£:1\)
IS
infinitely divisihle, and exactly the same argument shows more generally
.'
-160-
c(~B1'
that Cox processes directed by
for any
c
o.
>
are infinitely divisible
. .. , ~Bk)
By Lemma 8.6,
(~B1'
,~Bk)
...
is then itself
infinitely divisible, and the proof is complete.
We shall now prove a related but quite different criterion for infinite
divisibility which applies to the class of purely atomic random measures
with diffuse intensity.
~
define the measure
~
E
~E
and note that
Given any measure
E
E
B
B =
B
Band
E
the random measure
P{~B
THEOREM 11.5.
E~ E
E
> 0
and
B
discrete~ this
PROOF.
,E
Let
Then
M
d
E
0 , we
>
E
B ,
is discrete in the sense that it is purely atomic with
is measurable by Lemma 2.1.
S , and any
M and any E
E
M by
finitely many atoms in each compact set.
~ ~ ~E
~
~
~B,E
E
M}
~
E
>
0
on
= P{~
Given any random measure
P{~
with
S
E
Note also that the mapping
E
B = o}
statement
0 , we further define
M
I
~ B = o}
M EM.
E
be a purely atomic random measure on
P{~
on
as one with distribution
is infinitely divisible iff
B 7,n th
>
~
B = o}
E
> 0
.
remains true w1:th
-1
P
B,E
I
p~-l
In the case 7'Jhen
E
S
with
for every
~
&s a.
r,.
= 0
The necessity is an immediate consequence of Lemma 6.6 and the
fact that Poisson processes have independent increments.
To prove the
-161-
~
converse, let us first assume that
for every
B
B satisfying
E:
d
assume that
~
s
E~
(
B a.e.
and any
f
O} > 0
P{~B =
E:
For any such
-1
P~B,O
F
c
I
P~
-1
B we may
~ B ~ ~B 0 is independent of n B .
,
nB where
~B +
is a.s. discrete and that
For
we obtain
L[ (f)
"'5
(5)
and this holds simultaneously for all
base
G
set
C.
and all
For every fixed
B + {s}
n
such that
we have
nB s
d
-+
B except for points
E:
$
n
some
n
4C
~
s
S
E:
in some fixed
, consider a sequence
B ,B , .. ,
l 2
~B
p~-l
d
-+ ~
n
p{~
E
p~-l
s
We may thus conclude from
is infinitely divisible.
~
B = O}
>
we get in place of (5)
and so
G
E:
Hence by (5) and Lemma 5.1,
We now consider the general case al1d hence assume that
whenever
E~-null
is a.s. discrete and has no fixed atoms,
1 , and so
-+
, and we get
n
Theorem 11. 2 that
s
Since
O}
P{~B
B belonging to some countable
0
Writing
~
d
= t;BE
+
n
BE
p~;~
I
pt;-l
with independent terms,
-162-
IL~
s
(f) - Lt,;(f)L n
(f)
BES
I
Et,;BE(ds)
E~(ds)
2
<
=
2
+
- L~(f)
ILt,;BE (f)
E[~(ds);t,;cB=O]
~
=E-t,;(7'd-:--s-=-)=p"""{t,;=--=B-=-=-O'"}
-t,;f
E[e
;t,;cB=O]
~
E -~f
---'P"""{'-t,;---=-B=-'O"'""'}-- - .e
+
E
E
l-P{t,; B=O}
E(t,;-t,;E) (ds)
1
2 ---=E=-:"t,;-=(-=-ds"""""')-- P{t,; B= O}
E
+ 2
P{t,; B=O}
E
As before, this holds for all
for all
E
s
E in some sequence
outside some
tending to
E
belonging to some countable base
E(~ - t,;E)
X0
£:
(s)
set
~
0
C'
as
E
+
for
0
fixed atoms, so for fixed
Given arbi trary
p{[-E B
>
O}
<
n
s E S
<
E
-1
.
a.e.
~E
B with
and all
X0
Choosing
C, simultaneously
P{t,; B=O}
E
g (s) :: E(t,; E
~
E
Et,;,
say for
)
P{t,; B = O}
E
~
B -}
as
0
{s}
s 4 C u C' , we may therefore choose
(8n)-1
and then
Writing
nn
BEG
< n
f.
(
E
so small that
for the corresponding random measure
-1
n E N
f
<:
By Lemma 5.1 it follows that
0
(ds)/E~(ds)
outside some null-
s
nBES ' we then obtain
proving that
>
E ~ 0 , so
a.s. as
is clearly a.s. discrete and has no
E we have
Nand
€.
g (s)
(8n)
0
~ - ~E
Now
set
E~-null
E E E , it follows from this by dominated convergence that
Furthermore, every
so small that
G.
by dominated convergence.
non-decreasing in
g
I
rc
some
-163-
. fylng
'
satls
Lt;, Ln
Lt;,
=
p~-l
I p~-l
s
Ss .
'
, an d
agaln
s
infinite divisibility of
As b e f ore, t h"lS lmp l'les
and the proof is complete.
t;"
We are now going to consider extensions of Proposition 11.1 in another
direction.
THEOREM 11.6.
Let
6 )-1
P(~s s
some
tjl
.
be a point process on
~
independent of
1,fj
Tn this case
tjl'(O)
=
S
-1
-'
and
E:
M
Then
d
M(Et;"
iff t;,
d
M(H, -tjl')
t;,s - 6 s
Et;,
a.e.
S
E:
Et;,
lJith
S
fOY'
tjl)
a.e.
It is interesting to see how the specialization of Proposition 11.1
to point processes follows from this result simply by solving the differential
equation
=
tjl
subject to the initial condition
-tjl'
=
tjl(O)
1
The proof of Theorem 11.6 is based on the following lemma, giving
an interpretation of the notion of Palm distributions in terms of ordinary
conditional distributions.
LEr~MA
11. 7.
n
,
1AJ
I
n
C
Let
E.
~
B and
be a p01:nt process on
M
one of ihe atoms of
p{~
E.
M
I
N
E:
E,
For
&n
C
~C = n, T = s}
~C >
0
with
S
P{t;,
s
E:
M
E:
-'
and let
u)C fuy,theY' asswne that
chosen at random.
-'
EE;
MIt;,
s
c
Then
s ( C ,
n}
a. e. u)i th Y'eBpect to the measure
E[E:(ds);
PROOF.
The
n
~C
atoms in
nJ
= P{ E; C
s
n}EE;(ds)
s c S .
C being symmetric, we get for
see
T
-164-
P{E; E M, E;C
n, T E ds} = E[o (ds); t;, E M, E;C = n]
T
-1
= n E[t;,(ds); t;, E M, t;,C = n]
= n-lp{t;, s E M, t;, s C = n}Et;,(ds)
and in particular
P{t;,C = n,
T
E
ds}
By the chain rule for Radon-Nikodym derivatives, we thus obtain a.e.
p{E; E M
I
P{ t;, EM, t;, C=n}
E;C = n, T = s} = P{t;,EM, E;C=n, TEds}
P{ E;C=n, TEds}
= P{E;
s
s
s
P{ t;, C=n}
s
E MIt;, C = n} ,
s
as desired.
Proof of Theorem 11.6.
9,
E;B
d
Suppose that
t;, = M(Et;"
has then the generating function
¢) .
As shown in Section
¢(EE;B(l - t)) ,
t E [0,1] ,
and we get in particular
P{E;B = l} =
d~
¢(EE;B(l - t))
I t = °= -
Using this fact and Corollary 9.2, we obtian for any
fr:
P{ (E;. - 0 ) B = o} EE; (ds ) =
s
s
f P{E;
r:
B
B
EE;B ¢' (EE;B)
l}EE;(ds)
=
B .
E
B,C E B with
qr,C; r,B
B
c
1]
S
= P{E;B = l}E[~C
I
eB
1]
_. 1;[,13
,j,'
((;[,13)
'I'
- cf>'(EE;B)Ef,C .
On the other hand we get hy Corollary 9.2 for any disjoint
B,C E B
H.C
1:F;B
C
-165-
I
C
P{(I:
S
O}EI:(ds) =
-8)B
5
Ic
P{I: B
O}Eqds)
5
E[I:C; I:B
0]
00
I
nP{I:C
n,
E;B
= O}
n=l
00
=
I
I
np{qB u C) = n} P{ E;C = n
n=l
00
=
I
nP{t;(B u C)
n} { EE; HC
(BUC)
n=l
00
EE;C
d
EqBuC) dt
\'
L.
n=O
E;(B u C)
n}
r
n
t P{E;(B u C) = n}
EE;C
( (B u C) (1 - t) )
= EqBuC)
dtd <PEE;
It
EE;Cjl:[, (sue)
I
t = EE;CjEE;(BuC)
EEC
(
EE;(BuC)
EE;(B u C) <P'EE;(B
u C) - EE;C )
- - <P' (EEB) EE;C
These calculations show that for any
P{(~
If
5
B
E
B
- 8 )B = O} = - <P' (EEB)
5 E S
5
EE; c M
d ' then
by Theorem 3.3 that
[,
and hence also
E;
s
E5 - 8 5 ~ M(EE; ,-<P')
.
-
8
s
a.e.
EE;.
is simple, an<1it follows
Tn the general case we have
to evaluate
n l'
...
, (E;
s - 8 5 ) Bk
keN
But these probabilities arc determined hy the expectations
-166-
E[s(ds); sB
= nl ,
l
... ,sBk
= nk ]
, so they are uniquely determined by
the finite-dimensional distributions of
and
~
only.
Conversely, suppose that
Es
c
n c N be fixed with
as in Lemma 11.7, we get for
P{(s - 0,) Bl
k ,
l
:=
...
d
some
s
n
kl ,
and let
s E B
l
a.e.
, (s - o, ) Bm := k m
k
:=
= P{(s s
= P{nB l
P{E; - 0
,
:=
n] .
E M
proving that
,
r,C = n .
T
in
If
I
I
sC
:=
ssB 2
k ,
l
]
...
I
nC
,k
m
n,
:=
m
,
::
:=
k
, ssBm
os ) Bm :=
:=
B
E
B be an
E
E Z
E[E;(ds); sC
...
, (ss -
, nB m = km
Defining
+
n]
:=
s}
m
:=
I
sC
k
I
m
k
m
I
:=
n, , = s}
s sC
(ss -
n}
:=
os )C
:=
n - l}
n - l} ,
B , ... ,Bm ' this remains true for
l
sC
=
,
n, ,
s - 0
"
n
s}
,
:=
SEC
a.e.
for every
P{n
E
M
I
nC
n -
l}
are conditionally independent, given that
is a random cnumeration of the atom positions
C , we may easily conclude that
h.,i~j}
k ,
2
:=
...
m
C
Let
Es .
By Dynkin's theorem, we can extend this result to
and
l
+ 1,
k ,
l
and by the symmetry in
E[E;(ds); E;C
1
os) Bl
:=
a.e.
,B
, sB
= P{ ssB l
Md or not, and
Es E
n} > 0 , let
C
Es
must be generally valid.
0
-
"'s
arbitrary disjoint partition of
T
which in turn depend on
Thus it makes no difference whether
the above conclusion for diffuse
and
s
,.
J
j E {l, ... ,n}
is conditionally independent of
But then
must
Ilc conditionally independent, and since they arc also equally distributcd,
-167-
Ci;
B
is conditionally a sample process.
E
B with
Ph 1 ( B
I
B
c
oW
C
E[o
n}
i;C
Furthermore, we have for any
B
I
i;C = n]
n
'1
-lE[eB
,s
I
eC = ] = E[i;B; i;C=n]
,n
n P{ i;C =n J
s
IBP{i;sc=nJEi;(dS) = ]BP{(i;s-os)c=n-1H:i;(dS)
n P{i;C=n}
n P{i;C=n}
=
P{nC=n-l}Ei;B = Ei;Bfcp{(i;s-os)c=n-l}Ei;(dS)
n P{i;C=n}
n Ei;C P{i;C=n}
Ei;B
Ei;B
n Ei;C E[i;C; i;C=n] = Ei;C '
= Ei;B E[SC; i;C=n]
n Ei;C P{i;C=n}
,.
showing that the common conditional distribution of the
of
n.
It follows that
Ci;
J
is independent
is unconditionally a mixed sample process,
and an appl ication of Coroll ary 9.2 completes the proof.
We shall finally consider an extension of Theorem 11.6 to gener;l1
random measures.
IS
I{ecall that a measure is deycner'ate if its mass
confined to one single point.
S , then
is a random measure on
distributed w.r.t. some
for arbitrary
kEN
w
If
n
(n,s)
is a random variahle while
is said to be symmetrically
EM, if the distrihution of
and disjoint
1';
B , ... ,B E B
k
l
(n, sBl'
only depends on
For convenience, the random measures descri hed in parts
(i) and (jj) of Theorem 9.4 arc said to he of Type 1\ and B respectively.
We shall a 11 ow the measure
w
there to he fi n i te in both cases.
Note
that the two classes of random measures will then no longer he mutually
exclusive.
The canonical random elements
(lX,/-)
and
(a,())
:lrc tlefined
as in Theorem 9.4, and in case of Type 1\ random measures, we further
-168define the canonical random measure
A on
R+
by
A(dx) = aoO(dx) + x\(dx) ,
x z 0 .
THEOREM 11.8.
~
Let
~
s -
~
be a random measure on
d
s {s}o s ) = some
(n,s)
independently of
~)
(except possibly for a.s. degenerate
distributed
~.r.t.
E~
distributed
~.r.t.
E~,
~ith
S
In this case,
(n,s)
and furthermore,
n
and
E~
E
M
a.e.
s
~
Then
EF,
iff
is symmetrically
is also symmetrically
is independent of
s
iff
either
(i)
~
p
(ii)
F,
on
'l-S
~
0
of Type
A ~ith
A
= pM a.s. for some random vaY'1:ab le
and some non-random
is of Type B ~ith
a
=
ME
0
and
~~
(R )
+
S
,
or
a mixed sample process
R'+
Note by comparison with (i) that the point process
also allowed to be a mixed Poisson process.
take
a
S
in (ii) is
In this case we can even
non-zero, proportional to the mixing variable.
To save space, we omit the rather complicated proof of this theorem
and refer the reader to Kallenherg (1974b).
NOTES.
Proposition 11.1 was proved for point processes by Kerstan and
Matthes (1964) and Mecke (1967), but the prescnt proof is adapted from
.lagers (1973).
(1~)67),
Theorem 11.2 is clue to Kerstan and Matthes
(1~)64),
Mecke
Kummer ancl Matthes (1970) and others (cf. Kerstan,
~1atthes
and
Mecke (1974), page 116), while the extensions given in Theorems 11.3 and
-16911.4 are new.
Theorem 11.5 extends a result for point processes, proved
by Matthes (1969) with entirely different methods, (cf. Kerstan, Matthes
and Mecke (1974), page 97).
Finally, Theorem 11.6 is due to Slivnyak
(1962), Papangelou (1974) and Kallenberg (1973a), while Theorem 11.8 was
given in Kallenberg (1974 b ).
PROBLEMS.
11.1.
r;
Let
be a point process on
be a DC-semiring.
S
Show that
t;
with
Et;
M and let
E:
I
13
c
is infinitely divisible iff
k
for every function
f
I
=
II
j =1
II' ... ,I
11.2.
Suppose that
function
with
11.3.
t;S
f
<
E:
P(t;f) -1
Let
t;
Pit;
Pit;
~
I
O}
>
=
a
~
F
such that, given any point process
t;
a}
and
= O.
pt;-l
Show that there exists some fixed
is infinitely divisible iff
P(t;ff)-l
S
and let
Show that
pr;;l
E,
P{t;
on
S
=
O}
0
>
use the result in Problem 6.4.)
(Hint:
I
N ,
E:
0 •
is countable.
be a point process on
0, E,B
d
S
a.s.,
00
and
t;f
I , such that
E:
k
k
'
j
E:
13
be such that
is infinitely divisible iff
E,
f
B
=
lB
(Hint: argue with
r;
as with the Cox processes in the proof of Theorem 11.4, and note
that the counterpart of (4) holds trivially in the present case.)
11.4.
Show that the condition
hy the assumption that
Er;;
EF,
E:
Md in Theorem 11.5 can he replaced
be a-finite and diffuse.
-170-
11.5.
Let
l
~
be a random measure on
be a random element in
f: M x S -)- R
+
I
E[f(~,T)
~S
~S <
E
a.s., and let
00
Show that, for
~ ~
0
u
0
>
has the conS
E
S
and
,
= U, T = s] = E[f(~s' s)
E[~(ds); ~S
holds a.e.
with
which for given
~/~S
ditional distribution
measurable
S
S
I
~sS
= u]
du] , (cf. Kallenberg (1974b)).
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Verallgemeinerung cines Satzes
Kerstan, J. , Matthes, K. and Mecke, J.
Punktprozessc.
(32)
Rev. Roumaine Math.
Kerstan, J. and Matthes, K.
lIin~in
(1964).
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Complctely random mC:Jsures.
l'acific
59-78.
Kolmogorov, A. N. and Fomin, S. V.
Englewood Cliffs, Prentice-Hall.
(1970).
Introductory Re:Jl
i\naly:.;i~.
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Kummer, G. and Matthes, K.
(1970).
Vera11gemeinerung eines Satzes
Rev. Roumaine Math. Pures Appl.
von Sliwnjak II-III.
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845-870,
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1631-1642.
(35)
Kurtz, T.
(1973).
Point Processes and completely monotone set functions.
Tech. report, Math. dept., Madison, Wisconsin.
(36)
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(1972).
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Proc. 6th Berkeley symp. math. statist. probab.
449-462.
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Lee, P. M.
(1967).
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Z. WahY'scheinlichkeitstheoY'ic VCr'l,). Geli1:ete.
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Lo~ve,
M.
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7.
147-160.
3rd ed.
Princeton, Van
Nostrand.
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Lukacs, E.
(1970).
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2nd ed.
London.
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Matthes, K.
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Unbeschrankt teilbare Verteilungsgesetze stationorere
zuHitl iger Punktfolgen.
Wise.
z.
Hochsch. ElektY'o. Ilmenau.
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235-238.
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Matthes, K.
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Eine characterisierung der kontinuerlichen
unbegrenzt tei1baren Verteilungsgesetze zufalliger Punktfolgen.
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Stationare zufall ige Mosse auf lokalkompakten
7.. WahNlcheinUchke1:tsthcoY'ic ver'l,). Gehiete.
Abe1schen Gruppen.
9.36-58.
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Mecke, J.
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(tl4)
(;elJ1: (::I;e .
Mecke, .J.
W'iumen.
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Monch, G.
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doppclt
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(,'ehieLf'.
10.
27~)-2R:i.
Some remarks on a property of the Poi sson
Sankhya~ Scr. 1\, 35, 29-34.
-175-
AUTHOR INDEX
41
4
Be1yaev
29
Bi 11 ings ley
4 40 43 44
100 101 104
BUh1mann
134
Daley
29
Davidson
134
Feller
86
Fomin
6
Gihman
29
Goldman
98
III
Grandell
37
55
Grige lioni s
98
Harris
37
55
Hin~in
98
149
Jagers
55
86
Kallenberg
1 17 29 37
136 149 150
Karbe
65
Kerstan
1
Kingman
98
Ko1mogorov
6
Kummer
37
111
Kurtz
55
65
Leadbetter
29
Lee
86
Lo~vc
12
38
Lukacs
87
112
M..'ltthes
1
17
37
38
65
86
111
134
135
149
150
16R
169
Mccke
1
17
37
38
65
86
III
134
ns
149
ISO
168
16~)
Monch
37
Nawrotzki
III
Ososkov
98
9
45
..
Bauer
137
111
47 51 52 54 55
109 142 143 145
62 65
149
91
72
116
17
65
98
38
149
65
134
168
55 56 67 86
168 169 170
86
III
134
98
135
III
149
134
150
135
168
169
168
149
62
150
73
-176Author Index (cont.)
Palm
98
Papange10u
149
Prohorov
37
SS
Renyi
37
III
Ryll-Nardzewski
149
Sierpinski
9
Simmons
63
Skorohod
29
Slivnyak
149
Wa1denfels
37
Westcott
13S
149
169
85
169
S5
65
138
-177-
DEFINITION INDEX
bounded set
6
Campbell measure
136
Campbell's theorem
138
canonical rantlom clements
120
completely monotone
63
compound
17
covering
7
Cox process
16
DC-ring, DC-semiring
6
degenerate
87
diffuse
18
discrete
160
tlistribution
11
Dynkin's theorem
9
intlepcntlent increments
87
infinitely divisible
67
intensity
11
L-· transform
11
mixed Poisson process
16
mixed sample process
14
mixing
12
nested partitions
2S
null-array of partitions
20
null-array of random measures
66
Palm distribution
137
point process
10
Poisson process
16, 17
random measure
10
rcI!, lda r
24, 28, 48, 92, 93
sample process
14
simple point process
28
symllletrically distributed
113, 167
152
-179-
SYMBOL INDEX
,
p
6
aB
26
S
6
1;*
28
B
6
30
B
6
].1 V
d
F=F(S)
FC =F C (S)
M=M(S)
N=N(S)
6
Z
=
31
II . II
32
d
6
-+
39
7
BI;
40
7
~, ~
41
7
~
42
].1f
7
wd
~
52
M
7
t!.A
63
N
7
1T ,1T ,1T
f V
t
67
7
K
67
B
7
lims i
68
f].l
8
R'
68
Bp
8
E
£
68
1
8
I(a,A),I(A)
78
0(' )
9, 18
M(w,<P)
114
(rt,A,P)
10
PX,].1' QX,].1' Rx
129
11
Ps
136
11
!;s
+
C
B
--
13
PI;
-1
EE.:
+
138
~
I.e
11
!;f,l;f
140
0
14
f\
152
t~J
18
I
153
Po
18
ft;
155
\{'
19
lJE:
160
Z
19
E,
IbO
s
+
+
I•I
20
*,
20
1J'
20
r" k '"J" h)
22
11
f
s
13,£
-180-
INDEX OF SOME SCATTERED TOPICS
atomic properties
18ff, 56, 80, 99
canonical representations
68ff, 89, 120, 133ff
compound point processes
17, 32, 36, 99ff, 112, 135
convergence in distribution
39ff, 57f, 67ff, 87ff, 99ff, 142ff
Cox processes
16, 30, 32, 38, 56, 105, 109ff, 125, 157f
independent increments
87ff, 105, 119
infinite divisibility
67ff, 87ff, 112, 127ff, 152ff
mixed Poisson and sample proc.
14, 16, 113ff, 163, 168
null-arrays of random measures
66ff, 89ff, 147
Palm distributions
136ff, 151ff
Poisson processes
16f, 78, 91f, 118,147,151
sample processes
14, 118, 167
simplicity and diffuseness
28, 32ff, 48ff, 58, 64, 93ff, 114ff
symmetric distributions
113ff, 167f
tightness criteria
44, 70
thinned point processes
17, 30, 34, 56, 100, 109ff, 112, 125, 136, 158
uniqueness
30ff, 93, 105, 141