lResearch sponsored in part by the Office of Naval Research under
Contract N00014-67A-032l-0002, Task NR042214 with the University
of North Carolina.
2Replaces report No. 908, "A limit theorem for thinning of point processes".
LIMITS OF COMPOUND AND THINNED POINT PROCESSES 1
Olav Kallenberg
University of G~teborg, Sweden
University of North Carolina, Chapel Hill
Institute of Statistios ~meo Sepies No. 923
May 1974
,,'.
2
LIMITS OF COMPOUND AND THINNED POINT PROCESSES
Olav Kallenberg
Abstract
n
Let
= LOr
j
be a point process on some space
S
and let
j
6,61,6 , •.. be identically distributed non-negative random variables
2
which are mutually independent and independent of
~
form the compound point process
= L6 j Or
j
on
S.
of
~ as
1
which is a random measure
j
The purpose of this paper is to study the limiting behavior
and
6 ~ O.
In the particular case when
0 with probabilities
a p-thinning of
by
n. We can then
R~nyi
n
p
and I-p
6 takes the values
respectively,
~
becomes
and our theorems contain some classical results
and others on the thinnings of a fixed process, as well as
a characterization by Mecke of the class of subordinated Poisson processes.
COMPOUND AND THINNED POINT PROCESSES; INFINITELY DIVISIBLE RANDOM MEASURES;
SUBORDINATED POISSON PROCESSES; CONVERGENCE IN DISTRIBUTION; REGULARITY AND
DIFFUSENESS
1.
Introduction
Let
B be the ring of bounded Borel sets in S.
let
~
S be a locally compact second countable Hausdorff space and
on
By a mndom measure
S we shall mean a mapping of some probability space
(Q,A,p)
Research sponsored in part by the Office of Naval Research under Contract
N00014-67A-0321-0002, Task NR042214 with the University of North Carolina
-2-
M= M(s)
into the space
of Radon measures on
is a random variable for each
N
the subspace
B
B.
€
(SsB)
~
When
~B
such that
is a.s. confined to
M of integer-valued measures s it will also be aal1ed
c
a point ppoae88.
F for the class of measurable functions
Let us write
F
and
€
F
c
~ n ~ ~ s in
while 'Weak aonvergenae s
M s means that
on
+
for its subclass of continuous functions with bounded support.
c
Vague aonvepgenae s
f
S -+ R
~n ~ ~
and
M means that
~
w
-+ ~
n
~n S
-+
~ nf
-+
~f
f9r each
s defined for bounded measures
~S .
(Here
~f = J f(s)~(ds)
.)
Convergence in distribution [2] of random elements in topological spaces
~.
is written
taken to be the vague one.
It is known [11] that a sequence
random measures is tights and hence the sequence
distributions relatively compact [2]s iff
B
€
B.
M is
For random measures s the underlying topology in
Furthermore [4]s
{~ B}
n
{~n}
of
of corresponding
is tight for each
~n ~ ~ iff
(1.1 )
where
B~<"
= {B
€
B :
~aB = 0 a.s.} s and also iff ~ n f ~ ~fs f
In terms of L-transforms (L
measures
!;
by
L~(f)
= Lap1ace)s
= Ee-~f
s f
€
€
Fc
defined for arbitrary random
F s the latter criterion can be
written (cf. [11])
(1. 2)
It may be shown that any point process
n on S has the representation
-3where
is a Z+-valued random variable while
v
of random elements in
defined for
s
€
indicator of
S by
B.)
{T }
S without any limit point.
0sB = IB(s), B
Now suppose that
€
is a sequence
j
°s
(Here
€
N is
B ,where IB denotes the
6,8 1 ,6 , •.. are identically dis2
tributed R+-valued random variables which are mutually independent and
independent of
n , and define
v
= I
~
~B = 18jlB(Tj) < 00, B
Since
j=l
€
8j OT
j
s
B , it is easily seen that
is a random
j
measure on
S.
by
8, and we write
n
and
case when
and
l-p
8
We shall say that
~ ~ C(n,8)
only takes the values
respectively,
~
is a aompound point proaess determined
~
~
In the particular
0, with probabilities
will be called a p-thinning of
n
and
p
n .
{8 j } , the L-transform
may be calculated by means of Fubini's theorem, Le. we may first
n as non-random, and then perform mixing with respect to its
consider
distribution.
n
and
1
By the assumed independence of
of
for brevity.
= lJ = I(\
j
Writing
~ = La ' we obtain for
f E
f
and non-random
j
Ee-~f
=E
exP[-Iajf(t j )]
= ~E
= exp
exp[-ajf(t j )]
lj
log ~
0
= ~~
0
f(t j )
f(t ) = exp(lJ log ~
j
0
f) ,
and hence in general, by mixing,
f E
This shows in particular that
F •
p~-l does not depend on the representation
(1.4 )
-4(1.3) (which is not unique, even apart from the order of terms).
The
above mixing procedure also provides an alternative way of defining a
~ ~ C(n,S) , (cf. [7] page 359; use 1.6.2 in [6] to check measurability).
In the particular case when
~
is a p-thinning of
n , (1.4) takes the
form (cf. [8])
f
€
(1. 5)
F .
The main purpose of this paper is to study the limiting behavior
as
B~
of compound point processes.
0
We shall be able (in Section
3) to describe the class of possible limits, and under a mild regularity
condition (which is not needed in the thinning case), necessary and
sufficient conditions will be given for convergence in distribution
to a specified member of this class.
simple form for thinnings.
The results take a particularly
In this case, the class of limits consists
of all subordinated Poisson (SP-) processes (often called doubly stochastic
Poisson, or simply Cox processes).
nn
for each
n
directed by
€
N and if
n ) iff
p n
n n
Pn
~
If, moreover,
° , then
~n
is a Pn-thinning of
~ n ~ SP(n) (the SP-process
~ n . This proposition contains as particular
cases the classical thinning results by R~nyi (1956), Nawrotzki (1962),
Belyaev (1963) and Goldman (1967) (cf. Theorem 6.10.1 in [6]), who all
consider p-thinnings of some fixed point process and change the "time"
scale by the factor
P
-1
It also contains the following interesting
characterization by Mecke (1968) (cf. Theorem 5.6.12 in
class of SP-processes:
for each
p
€
(0,1].
[6]) of the
A point process in SP iff it is a p-thinning
-5A secondary result of some independent interest is Lemma 4 which
indicates how the classical regularity conditions ensuring a point
process to be simple (see e.g. Satz 1.3.5 in [6] and Theorem 2.5 in
[4]) have analogues ensuring an arbitrary random measure to be diffuse.
Even the converse proposition (Dobrushinls lemma) carries over to this
context.
2.
Preliminaries
In this section we shall introduce the class of limiting random
measures and prove some auxiliary results.
use
g
to denote the function
or
For any measure
l-e
Throughout the paper we
-x
R I we define a new measure
+
on
Our first lemma is essentially contained in Theorem 3.1 of [4]
(which needs correction: condition (i) must also hold with liminf instead
of limsup).
Lemma 1.
Let (l
E
R+
and
A E M(R~)
with (l + Ag
= 1" and define
RI(l - e-tx)A(dx),
+
Further suppose that
(2.1)
81 ,8 2 , ... are R+-valued random variables such
that 8n ~ 0 " and put
~n
= L 8n ,
cn
= Eg(8 n ) = 1
-
~
n
(1),
n
EN.
~n ~ ~ unifo~ly on bounded intervals iff
Then - c-llog
n
( -l)
c n-1 gPa
n
w
~
(l00 + .gA •
-6Let
measure
a
and
A be such as in the lemma and consider an arbitrary
~ €
M.
Then the measures
M
a~ €
and
Ax
~ E M(R~ x
S)
may serve as the canonical measures of an infinitely divisible random
~
measure
S with (~-homogeneous) independent increments (see
on
e.g. [4]), the L-transform of which is given for
= a~f
_t"f
- log
Ee"
+
fR~XS(l
= fs{af(t)
Again we may consider
=n
~
+
-
f
e-Xf(t))(~
fR~(l
-
€
F by
~)(dxdt)
x
e-Xf(t))A(dx)}~(dt) = ~(1/1
0
f) .
as a random measure and mix with respect
to its distribution (check the measurability by means of 1.6.2 in [6]),
to obtain
f
The distribution of
Write for brevity
1/1
obtain
A=
°1
,
say that
=g
being determined by
~
~ ~ S(n,a,A), (S
~
and
d
=
SP( n)
-1
, a
F •
and
n
is non-zero in
Note that we
in the particular case when
Lemma 2.
If
~
C if
nC ~ 0
a
=0
C
For
and diffuse there is
and
B
E
n
C .
d
= S(n,a,A) , where
to be non-aero and diffuse in some region
are uniqueZy
(2.2)
A, we shall
for subordination).
We shall need the following uniqueness result.
has a.s. no atoms in
It.
Pn
€
dete~ined
by p~-l
a + Ag = 1
C
€
whiZe
B , then
Pn
is known
n
-1
.J
a
and
.
The proof of Lemma 2 is based on the following result which is
obtained in the same way as Satz
of 1.3.10.
5.6.9 in [6] by using 1.3.7 in place
-7-
~8
Let
n be a diffu8e mndom measure on S
uniquely detePmined by the quantities Ee -nB ,B E B •
Lemma 3.
Proof of Lemma 2.
B
nC
nC ~
° ,LnC
proving the uniqueness of
-1
= LnC
II· /I
that
on
(0,1]
$
0
(tf)),
f
E
F,
If
Ln(f)
t
E
f
$
E
Pn
-1
f
,since
€
-1
on
F is such
= Ls ($-1
o
f) , so
I II/J " ,
[O'WlT J
B.Y the uniqueness of analytic continuations it follows that
• unique, and hence so is
A
and
a
has a unique inverse
denotes the supremum norm.
= Ee -tnf = Ls ($-l
and we obtain
LsC '
0
Ilfll < 11$11 , we thus obtain from (2.2)
Lnf(t)
C , and in
$(t),
$ , and hence by Lemma 1 of
Now it may be seen from (2.1) that
[0,/1$/1) , where
0
has a unique inverse
$
n to
From (2.2) we fUrther obtain
is unique.
= Ee-tsC = Ee-nC$(t) = L
Since
Ps-l
B here by B n C , it follows by Lemma 3 that
p(nc)-l
-1
B•
E
determines the distribution of the restriction of
particular
Pn
B.Y (2.2),
Ee -nB ,
If we replace
Then
•
is
F was arbitrary.
To ensure diffuseness, we introduce regUlarity conditions as follows.
Let us say that
{Cmj}
c
n
is regular in
B of finite partitions of
C
€
B if there exists some array
C (one for each m) such that
-8lim
L P{nCmj
= 0,
> €}
> 0 •
€
rr.t+co j
More generally, if
say that
n,n ,n ,
l 2
are random measures on
is n-regula1' in
{nn}
C
Bn if there exists some array
E
Bn of partitions of C such that
{C j} c
m
lim limsup L P{n Cmj > €}
n-+<» j
n
= 0,
rr.t+co
4. Let n,n l ,n 2 ,
Lemma
and suppose that
in
E
B
if EnC
<
C
00
n
Then
n
Proof.
.••
is regula1' in
C
n is diffuse in
> €} S
j
max
j
ICmj I -+
B
C.
mj
j
L liminf
j
n-+<»
{nn}
01'
that
S
{nn}
with
is
n-regular
The converses are also true
C and
0 ,
(I. -,
n
follows from the relation
>d.
> €}
with
P{n Cmj > €} S limsup L p{n C j > €},
n
n-+<» j
n m
is regular and hence diffuse.
diffuse on
E
The diffuseness of a regular
We further obtain for n-regular
n
0 .
•
SEC
L P{nCmj
€ >
be random measures on
p{max n{s} > €} S p{max nC
so
S, we shall
EnC <
00
,
Conversely, suppose that
and let us choose
denoting the diameter).
{Cmj}
c
Then clearly max nCmj
a. s., so
€Lj
l{ C >}
n mj e
-+
0
a. s. ,
n
€
> 0 •
and hence by dominated convergence
€
> 0 ,
0 ,
is
B such that
j
nC ~
£ >
-+ 0
-9proving regularity of
n
If we choose the
Cmj
in
B (which is
n
always possible, cf. the Remark in [4] page 10), we further obtain
limsup L p{n C j > €} ~ L limsup P{nnCmj ~
n-+«> j
n m
j
n-+«>
and so
3.
{nn}
is n-regular.
Main results
Recall the definitions of
and
1
a + Ag
2
respectively.
C(n,B)
and
S(n,a,A) , given in Sections
We shall always assume that
B~ 0
and that
= 1.
Theorem 1.
For eaah n
Bn ~ o.
Suppose that
€
~n ~
Then
~n g C(nn,Bn ) and put
N , let
c
n
= Eg(Bn )
iff cnnn ~ 0 • Furthe1'f7lore, the
0
aonditions
c n
n n
(ii)
imply that
~ n ,
c -1 g ( PB -1) w
+ ao + gA
O
n
n
~n ~ ~ implies that ~ ~
~ n ~ S(n,a,A) • Conversely,
some S(n,a,A) , and if {c n nn } is ~-regular in some C € B~ with
~C ~ o , then (i) and (ii) are satisfied for some n , a and A .
No regularity condition is needed in the thinning case:
Theorem 2.
a
p
n
For eaah n
€
N , let
-thinning of some point proaess
~ n ~ some ~ iff
p
n nn
Pn
€
(0,1]
~n
and let
nn • Suppose that
p
n
+
be
0 .
~ some n , and in this aase ~ ~ SP(n) •
Then
.
-10Corollary (Mecke).
Bome
iff
SP(n)
RemaPk.
~
Let
~ be a point process on S.
~
Then
is distributed as a p-thinning fop each
p E
g
(0,1] .
As was pointed out in [5], the last result is essentially
contained in Theorems 5.1 and 5.2 of that paper.
It may be of some
interest to observe that even Theorem 2 above (in particular cases,
S
such as when
= R+)
2.3 and 4.2 there).
4.
can be obtained from results in [5], (viz. Theorems
We leave details to the reader
Proofs
Proof of Theorem 1.
~ n ~ 0 iff c n nn ~ 0 , note
To prove that
that by (1. 4)
. -t~ B
Ee
n
where
<p
=E
= LS
n-
exp[nnB log $n(t)],
t
E
R+, B
E
B,
(4.1)
n EN,
Using the elementary inequalities
n
1 - x ~ - log x ~ 2(1 - x),
and the fact that
<p
n
since
-+ 1
(4.2)
Sn ~ 0 , we get for sufficiently large n
2c
so by (4.1) with t
=1
-~ B
E exp(- 2c n B) ~ Ee' n ~ E exp(- c n B),
n n
n n
Letting
n
-+ ~ ,
n
it is seen that
B
E
B•
~ nB ~ 0 iff c nnnB ~ 0 , B
E
B ,
and the assertion follows.
Let us next suppose that
that in this case
~n ~ ~
(i)
= S(n,a,A}
-
and
(li)
are satisfied.
To show
, it suffices to verify (1.2),
E
N
-11i.e. by (1.4) and (2.2) to show that, for any fixed
E exp(n log ~
n
where
~
is defined by (2.1).
and continuous on
n
f €
F
c
f) ~ Ee-n(~of) ,
0
Since the function
e -x
is bounded
R+ ' it is enough by Theorem 5.2 in [2] to show that
and by Theorem 5.5 in [2] it suffices to prove this for non-random
satisfying
€
(i) , or more generally, to show that for any
M with
II
v
~
n
II
(4.3)
Since
f
~(O)
has compact support and
we may assume that even
II
n
¥
II ,
=0
, ~1(0)
= ~2(0) = ... = 1
and after normalization (Which is
= 0)
possible except in the trivial case
llS
probability measures.
(4.3) may be written in the form
- c
In this case,
-1
n
~
E log
0
n
,
f(T n )
~ E~
1l'1l1'1l2' ••• are
,that
0
f(T) ,
T,Tl ,T2 , ••• are random elements in S with distributions
1l,1l1,1l2' ••• , hence satisfying Tn ~ T. By (ii) and Lemma 1, the
where
sequence
f
{c~llog ~n} is uniformly bounded on finite intervals, and
being bounded, it follows that {c
-1
n
log
~
~
log
n
0
f}
(4.4) is implied by
Hence, by Theorem 5.2 in [2],
- c
-1
n
n
0
n
f(T n ) +
~
0
f(T) .
is uniformly bounded.
(4.4)
-12Applying Theorem 5.5 in [2] once more, it is seen that (4.5) needs verification only for non-random T,T ,T , .•. with
l 2
Tn
~
T , and so,
f
being
continuous, it remains to prove that
-1
- c n log ~ n (xn ) ~ ~(x)
x
F;
(ii) , and so the convergence
from
Suppose conversely that
F;
n
~
n
n
x .
But by Lemma 1 this follows
~ F;
is established.
~ some
F; , and let
B
E
BF;
be arbitrary.
F; B ~ F;B (cf. (1.1)), so by (4.1)
Then
n
E exp[n B log ~ (t)] ~ Ee-tF;B,
n
n
and since this limit tends to one as
£
>
0
some
t
E
(0,1)
t
~
0 , there exists for each
satisfying
E exp[n nB log
~
n (t)] ~ 1 - £/2,
(4.6)
n EN.
Now it follows by (4.2) and the elementary inequality
1 - e
-tx
x
~ t(l - e-),
t
E
[0,1],
x
E
R+ '
that
- log ~ n (t) ~ 1 - ~n
(t) '= E(l _ e
-t8
n) ~ tE(l _ e
-8
n)
= tc n ,
n EN,
and so by (4.6) and ~eby~ev's inequality
p{c n B > t-llog 2}
nn
= P{l
= p{tc nn
n B >
log 2} ~ P{-n B log ~ (t) > log 2}
n
n
- exp[n B log ~ (t)] > 12} ~ 2E{1 - exp[n B log ~ (t)]} ~ £ ,
n
n
n
n
-13proving tightness of
that
{c n}
{c
n B}
n n
Since
B
E B~
was arbitrarYt if follows
is tight.
n n
~n
Let us now assume that
i
~ ~
and prove that then
0
lim limsup c-l(l - ~ (t))
t-+O
n-+a>
n
n
=0
.
Suppose on the contrary that the limit in (4.7) is greater than some
E > O.
sequence
N'
c
and since
{c n }
n n
for which
cnn n
nB ~ 0 •
-t.; B
n
E
N'
(4.8)
t
is tight t we may choose some subsequence
i
some
n
.
Nil c N'
According to the first assertion t
n ~ o t so we may further choose some
implies
such that t for some
N t
n
Ee
t > 0
Then there exist arbitratily small
B E B~ n B
n
~
~ 0
with
By (4.l)t (4.2) and (4.8)t
=E
exp[n B log ~ (t)] ~ E exp[-n B(l - ~ (t))] ~ E exp(-ec n B)
n
n
n
n
n n
.; B ~ ';B and c n B + nB on Nil t i t follows that
n n
n
Ee -t';B <- Ee -EnB . But since t could be chosen arbitrarily small t
d
and since
this yields the contradiction
1 ~ Ee-
enB
< 1 t proving that (4.7) is
indeed true.
By ~eby~ev's inequality we further obtain for any
c-l p{13
n
rtt > 0
n
and letting in order
n + ~t r + ~
and
t + 0 t we get by (4.7)
-14lim limsup c-lp{S
n
n-+co
r+a>
o-l)}
{ c -1 g ( P~n
which shows that the sequence
on
R+
is tight.
and some
n,a
=0
r}
>
n
,
of probability measures
n
In particular there exist some sequence
and
A such that
(i)
and
(ii)
hold on
N'
N' , and
we may conclude from the sUfficiency part of the theorem that
the asserted form.
arbitrary
a
and
l )}
{C -lg(po~
n
subsequence
A.
C
Nil
n n
2.
~n
d
+
such that
BI;
(i)
= Bn
is n-regular on
n n
and choose
is
~-regular
N' c N must contain some
(ii) hold on Nil for some n,a
and
by (2.2), proving that
(i)
and
I;
d
=
S(n,a,A)
nC ~ 0
Pn
(ii)
-1
, a
,
and that
Using Lemma 4, it follows that
C
C , and therefore
This proves
=0
, and that . {c n }
~.
By the direct part of the theorem we have
diffuse in
n
and
B~ with ~C ~ 0 • Since the sequences {cn n}
n
E
and in particular
{c n}
we may take
are .both tight, any sequence
n
and
go,
has
~
A.)
Let us finally assume that
in some region
~
(In the case
N
c
n
is
are unique by Lemma
and
for the original sequence (cf. Theorem
2.3 in [2]), and the proof is complete.
Proof of Theorem 2.
a
=0
since
and
The preceding proof applies with
c
= pn
n -
,
A = 01 ' (and it may even be simplified in the present case,
(ii)
is automatically satisfied).
needed, since if
I;
g
Pn-l
SP(n) , then
No regularity assumption is
is uniquely determined by
PI;-l , (cf. [6] page 316 or the proof of Lemma 2 above).
Proof of the Corollary.
be arbitrary.
Let
I;
g
and if
some
SP(n)
and let
p
is a p-thinning of
E
(0,1]
n ,
p
-15we get by (l.5) and (2.2) for any
f
E
L~ (f)
p
F
L -1 (1 - [1 - p(l - ep
f
-1 (p(l_e- })
P n
=L
and so
~
d
=
~
.
p
This proves that
Conversely, suppose that
Pn
= n -1
and
~n
=~ ,
n
~
f
)]}
n
= Ln(l-e- f } = L~(f}
is a p-thinning for each
~
has this property.
€
N , it is seen that
p
E
(0,1].
Applying Theorem 2 with
~
must be an SP-process.
References
[1]
BELYAEV, YU.K.
Theor. Probab. Appl.
[2]
~,
(1963) Limit theorems for dissipative flows.
165 - 173.
BILLINGSLEY, P. (1968)
Convergence of probability measures.
Wiley, New York.
[3]
GOLDMAN, J. R. (1967) Stochastic point processes: limit theorems.
Ann. Math. Statist. 38, 771 - 779.
[4]
KALLENBERG, O. (1973)
Characterization and convergence of
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Gebiete 27, 9 - 21.
[5]
KALLENBERG, O.
(1973) Canonical representations and convergence
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[6]
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[7]
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[8]
MECKE, J. (1968) Eine characteristische Eigenschaft der doppelt
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[9]
NAWROTZKI, K. (1962) Ein Grenzwertsatz fUr homogene zufgllige
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Math. Naahr.
24, 201 - 217.
[10]
HENYI, A. (1956) A characterization of Poisson processes.
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Mat.
Kutat~
[11]
MaSe.
Int.
K~3L.
l,
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