'e
This wor-k was suppol'ted by the Offloe of NavaZ Research
N00014-6?-A-OD2t-0002.
II
STATIONARY STOCHASTIC POINT PROCESSES
by Johannes Ker-stan and KZaus Matthes
STATIONARY STOCHASTIC PoINT PROCESSES
III
by P. Fr-anken~ A. Liemant and K. Matthes
Translated by
D. R. Daley and C. E. Jeffcoat
Departtment of Statistics
Univer-sity of Nor-th Car-oZina at ChapeZ Bin
Institute of Statistics Mimeo Series No. 775
Augus t, 1971
~!"~Epact
STATIONARY STOCHASTIC POINT PROCESSES
by
II
Johannes Ke1'stan and KZaus Matthes
STATIONARY STOCHASTI C POI NT PROCESSES II I
by
P. F1'anken, A. Liemant and K. Matthes
Translated by
D. R. Daley
C. E. Jeffcoat
This 'Wo1'k 'Was Suppo1'ted by the Offiae of NavaZ Resea1'ah unde1' aont1'aat
N00014-67-A-0321-0002.
ii
Prefatory Remarks Concerning This Translation
This paper is a translation from German of the second and.third of the
ries "Stationary stochastic p<>int 'prpcesses I, II and III",
tively by
se~
written respec-
K. Matthes; J. Kerstan and K. Matthes; P. Franken, A. Liemant and
K. Matthes in Deutsohe Math. Vereinigung,
the three papers appears
be1ow~)
Vols 66, 67.
(A list of contents of
Part I has been translated previously by
R. J. Serfling (Statistics Report Ml90, Florida State University).
The present translation of Parts II and III was done by C. E. Jeffcoat,
University of North Carolina, and D. J. Daley of the Australian National
University.
In particular, Dr. Daley has added footnotes where clarification
seemed desirable.
The notation of Serf1ing has been used - in replacing
Gothic letters by script.
It should be remarked that these papers are of unquestionable importance
in the Point Process literature.
They treat a point process in the main as a
special case of a marked point process.
This gives a very useful and general
framework, which does, of course, involve some notational complexity due to the
"mark" structure.
This notation can be simplified when marks are not relevant.
The numbering of the references and chapters carries throughout the threepart series, which is comprised of the following chapters:
Part I. (by K. Matthes) 66, 66-79.
1.
2.
3.
Foundations
Intensity and Palm distribution
Stationary Poisson point processes.
Part II. (by J. Kerstan and K. Matthes) 66, 106-118.
4.
5.
Stationary infinitely divisible distributions
Regular and singular infinitely divisible distributions.
iii
.
.
Part III. (by P.
Fpanken~
-
(~
A. Liemant and K. Matthes) 67, 183-202.
,...,
6.
7.
A new approach to PA,Q
Palm distributions in a generalized sense
8.
The distributions
9.
Generalization of a result of Palm and Khinchin
10.
Limit distributions for infinitesimal arrays of stationary renewal
processes
11.
Some relations between
(P)o
P
and
PL.
M. R. Leadbetter
STATI ONARY STOCHASTI C POI NT PROCESSES,
II
*
Johannes Kerstan and Klaus Matthes
4.
Let
~K
Stationary infinitely divisible distributions
MK
denote the smallest a-algebra of subsets of
1
to which the mapping
contained in
[-k,k)
~ ---~ N(~n(IxL»
and all
is measurable for all intervals
K.
L in
with respect
We say that
[K,K]
I
has Property (aJ
when the following is true:
If a finite measure
(a)
kU is defined on each
then there exists a com-
mon extension of the
U on
to a measure
kU'S
kU's
= Rn
U~=l kMK
MK•
satisfied if
K
MK•
because the algebra
Property (a) imposes no severe restrictions on
if
such that k"U is
an extension of k ,U whenever k" > k' -'
U is uniquely determined by the
generates
kMK
K
and
It is certainly
K is the family of all subsets of
is countable and
K
[K,K].
consists of the Borel subsets of
K,
or
n
R.
In this and the three subsequent chapters, we assume there is given a
[K,K]
fixed mark space
with Property (a).
We define a stationary distri-
MK to be infiniteZy divisibZe if there exist finite sequences
P l""'P
of stationary distributions on MK such that
n,
n,sn
bution
(1)
P
on
(PI
n,
* ... * Pn , s n )
-->
P
Stationare zufa11ige Punktfo1gen, II. Jber. Deutsch. Math.-Verein. 66
(1964) 106-118. Part I in same J. 66 (1963) 66-79.
1
Always here,
N(o)
=
card(.)
= (German) An.
(0).
2
and
for all positive
(2)
n
;:?;
n
t
there exists
t, e
n
such that for all
t,e
,e: '
.(~n«O,t)
P
x
n,~
K) #
~)
<
e.
From this definition it follows immediately that the convolution
of stationary infinitely divisible
divisible.
Also, if
exists
stationary with
Un
U
n
and
P
P
(~
PI' P
is stationary and for every positive integer
n
n «O,t)
= P,
K)
x
then for all positive
#~)
l
* P2
is also stationary infinitely
z
(U )n
P
0
+
n
there
t
(n + 00)
is infinitely divisible.
For all stationary distributions
the stationary distribution
EA,S
S
on
MK
and all
A > 0 we define
by
.m
~ Sm
m=O m!
00
= exp(-A)
L
Then
where we define
=
and, in particular,
(EA/n,S)
n
= EA,S.
Thus every
EA,S
is infinitely
divisible.
Observe that
EA,S
= o~
if and only if
S = o~.
For
S # S~
we have
by 1.2- the relation
=
We now give a simple model from which a second class of stationary infinitely divisible
P's
can be derived.
3
Let
Poisson distribution
bution
Q.
dent of
MK
Q be any distribution on
P
A
and let each
Further suppose the
~
n
's
and
A > O.
~n€MK
~€M'
Let
(-00 < n < 00)
have the
have the distri-
to be mutually independent and indepen-
set 2
<P.
x =
-00
In this way each
x.€~
l.
u
< i <
T
00
-x.
l.
defines a cluster (nSchauer ll )
T
~
-x. i
and
X
l.
consists of the superposition of all the clusters.
Clearly the mapping
L€K
[w,(~
and all bounded intervals
allowed.
n ) -oo<n<oo ]
I,
Our construction leads to
for all bounded
I, N(Xn(IxK»
<
00
~
N(Xn(IxL»
is measurable for all
where, of course, the value
+00
is
XEMK with probability one precisely when
with probability one.
On account of the
stationarity of P
1.
it is enough to verify this condition for some non-empty
A
Observe that then the distribution of X is stationary; denote it by
For every bounded interval
negative function of
bounded
I
x.
I,
Q(~n(I+x)
x K)
~~)
is a measurable non-
If it has a finite integral for some non-empty
then the same is true for every bounded interval.
By the Borel-
Cantelli lemma, it follows (cf. [27]) that
4.1.
PA,Q
Hence
i8 defined if and only if
Q(N(~)
For Q =
6~,
invariance property
< (0)
=1
is a necessary but not sufficient condition.
and only in this case,
3
J:oo Q(~n«I+x)xK) ~ ~)dx < co.
PA,Q
= o~.
On the basis of the first
of the Poisson distribution given in Chapter 3, we have the
2
Tt~
3
This is a reference to §3.3.
is defined in 1.1.
4
relation for
Q
~ o~
=
Q('¥~<P)
Now let
= 1.
of
'11* = T '11.
zl
Then with probability one
with
{[Zl,kll, •• ,,[zn,kn ]}
Zl < ••• < Zn •
°
=
~~)
On the basis of a sharper form (see
A
Let
AQ('¥~<P).
'11
has the form
denote the distribution
Q*
Then clearly
Q*('¥*n«-oo,O)xK)
P
=
A'
where
and
4
Q*('¥*n({O}xK) ~~)
=
1.
6.2) of the second invariance property of
given in Chapter 3, we have
=
We can, therefore, restrict the above model without loss of generality by requiring that,the clusters be subject to the causality principleS
observed reaction to an impulse occurring at time
Since
Thus every
Each
P
.
A1,Ql
PA,Q
EA,S
=P
*p
t.
/ '
can be represented as the limit of a weakly convergent se-
MK
tionary distribution on
'¥n«O,n)xK)
and
induced by
4.2.
PA,Q'
A > 0.
For let
The original has
MK.
Sn
the distri-
S, then
(n+
oivisible distribution on
S be an arbitrary sta-
If now we denote by
oo).
We now return to the general theory.
4
begin immediately at
A2 ,Q2
Al+A2,(AlQl+A2QZ) (Al+A2)
is infinitely divisible.
quence of distributions of the form
bution of
t
and that the
Let
P
bea stationa.ry infinitely
For all finite sequences
Q*('¥*n({O}xK)=~) =
6
! = {ll, ••• ,In },
1.
5
i.e., the Poisson process triggers a cluster in which all events (= points)
occur on or after the triggering event.
6
The notation
!
and 1
seems useful.
5
1
= {L1,·.·,Ln }
the distribution
Ii
of bounded interval$
Pr'L
..... ' .....
of the vector
and measurable subsets
[N(~n(IlxLI»,
•••
L
i
of
,N(~n(InxLn»]
K,
is in-
finitely divisible and therefore.
00
4.3. PI'L has the representation Pr'L = exp(-A)
",',.....,
..... ' .....
In 4.3,
°
A~
and
L
Am m
-, q •
m.
m=O
q. is the distribution of a random vector with non-nega-
tive integer-valued components.
p •••
, # 0[0 ,eoo, 0]
For
the requirement that
qr. T '
the representation 4.3 is uniquely determined by
q({[O, ••• ,O]}) = 0.
A = AI ' L and
..... ' .....
= 0 [0, ... ,0] then we set
If on the other hand
... 'Ii:l
q
We then set
A
=
q
=
° and
= 0 [0, ... ,0] •
Set
x
runs through all the vectors of
with non-negative integer-valued coordinates.
Then
p •••
,
n
R
is the convolution
of
<Xl
where
These
A
• ; ";X
are uniquely determined by
collectively determine
p •••
,
=
...,
A
Conversely the
P•
and therefore
A•••
, ,' x
P.
In the convolution of sta-
tionary infinitely divisible distributions the corresponding values of the
A0; ;x 's
0
are added.
From the
Ao;o;x 's
we now use an extension procedure common in measure
theory to derive a measure which will be of considerable assistance in studying
the structure of infinitely divisible distributions.
4.4.
To each stationary infinitely divisible distribution
co:rresponds
exactly one (possibZy infinite) measure
P
on M
K
on M
K
with the
P
properties
(a)
for aU finite sequences
L and I:: of bounded intervaZs and
6
measupable subsets of K respeatively and all non-null veators
m= [m1 ,···,mn ]
with non-negative integer-valued aoordinates
P(N{xn{IixL.»=m., i=l, ••• ,n) = A
1
(S)
!;k;m ; .
1
P({ep}) = 0;
The aorrespondence
P
+
'"
P gives a single-valued invertible mapping of the
class of all stationary infinitely divisible distributions on MK onto the
class of those measures on MK whiah are stationary relative to
addition to (S) have the property
(y)
for all bounded intervals
I
T
and in
t
the measure of {x: Xn{IxK) :f </J}
is finite.
Proof. Let H denote the totality of those subsets of MK that can be
{~: N{~n{IixLi»
put in the form H =
[0, ••• ,0].
= m , i = 1, ••• ,n}
i
H (cf. [26], §5)
The a-ring generated by
MK with the property
ep4X.
[m1 , ••• ,mn ]:f
with
contains every
In this proof, we call a function
H-finite1y additive if for all
Y on
X in
H
H in H
00
Y{H)
Let
L
k=O
Y({~: N(~n(I +l xL +1»
n
n
be a stationary distribution on
P
[N(~n(IixLi»'
tors
=
i=l, ••• ,n]
MK
P,
since the values on the a-ring gen-
H are uniquely determined by (a) and hence the values on all of
are uniquely determined because of (8).
nition of the
Ao;o;x 's
H-finite1y additive.
which
MK for which each of the vec-
has a joint distribution of the form 4.3.
Then there can be at most one measure
erated by
= k} n H).
I , ••• ,I
1
n
Let
It is easily seen from the defi-
that the equation in (a) leads to a set-function
kH
are contained in
exp{-A ••, )
X in
denote the ensemble of those
A•••
,
[-k,k).
For every
=
X in
, .>
exp ( - A••
kH,
A•••• m'
,
'",
H for
7
i.e.
exp(Ar.L)P(X) ~ P(X),
and hence a fortiori
-'-
exp(A ~k,k);K) P(X)
~ P(X).
On the basis of this inequality, it follows that the H-finite1y additive
set-function induced on
way to a measure
k"V
For
kMK
by means of (a) can be extended in exactly one
kV on the a-ring
is an extension of
measure on
kH
k'V,
kR
generated by
For each
m and all
kH.
k ll > k',
For
k > m we define a finite
by
k" > k' ,
of the mark space
is an extension of
the
[K,K]
kQm
k,Qm'
(k = m+1,m+2, ••. )
strictions of a well-defined finite measure
tone increasing in
m.
£lmm~up Qm
Thus
and «(3).
-P
Thus the existence of
Because of property (a)
may be regarded as re-
MK•
Q on
m
The
Qm are mono-
is a measure with the properties (a)
""
P
is established.
is clearly stationary and
because of (a) satisfies condition (y).
Let
H be defined as before with
x
=
[m , •• o,m ] # [0, ... ,0].
1
n
00
P(H)
=
ProL({X})
~~~
=
eXp(-A roL )
~,~
I
(A!;l?m
m=l
We now write this equation in another form.
Then
({x}) •
mt
Set
n
= p«o)n{x: I N(Xn(IiXL.» # oJ).
i=l
1
By using direct products, we can now form the convolution powers of the
finite measure
-P
as in the case of distributions.
({x})
=
Then for all
(H) •
m
~
1,
8
Thus
~~e
obtain (for
H as defined)·
Suppose now that
is a stationary measure on M satisfying both of
K
V
conditions (~) and (y).
additive set-function
~oo
P
Replacing
Z.
in (*) by
H€H,
For all
~
Z(H)
V yields an H-finitely
1.
If
H€kH,
then
m
1
P(H) ~ lm=l m! (V(-k,k);K) (H).
Thus for all
k
is an extension of
a measure
k'Z'.
= m+l,m+2, ••• )
The
ZI
m
Z'
on
We now set
and thus obtain for each
(k
k
kR
k > m a finite measure
JAK.
kZ'm on
may be regarded as the restrictions of some
measure on MK with the property
P(X)
=
{ZI(X)
Z'(X)
m,
Z '(~)
related to the given measure
~ 1.
~
P
and
V
ZI
MK•
is a
for
$ 4: X
for
$
MK
on
Then
t~e
€
X,
which for all
H€H
is
representation 4.3 is valid
~
= V.
The correspondence
one mapping of the set of those stationary
the set of all stationary
on
m
kZ'm
lve now set
+ (l-Z'(MK»
V by (*).
ZI
The
and their upper limit
and thus obtain a stationary distribution P
P,
k" > k I,
= kZ'«o)n{~n«-m,m)xK)+$})
kZ'm(o)
are monotonic increasing with
generally for
For
is given.
piS
P
~
P
thus gives a one-
for which 4.3 is valid onto
with the properties (S) and (y).
It remains to be shown that P is. infinitely divisible" if 4.3 holds generally.
For this we set
~-=-=..JJ
(p"*p
" 1
2
= ""p1+P""2'
1
and denote the associated distribution by
we have
(Q)
n
ever it follows that
4.4.
~
V=-P
P
n
n
= P.
From the existence of these
is infinitely divisibleo
Q.
n
Q
n
's
Since
how-
This completes the proof of
9
For
P(~)
<
A> 0
and
S(M'K)
Conversely, let
00.
= 1,
EA, S
P(=f: 0cP)
= AS.
be any stationary infinitely divisible
distribution with the property y:: P(~) <
P = E
S
==
A
~
and S(M'K) = 1 if and only if P
A> 0
and
S
ape unique Zy dete:mrined by
=
P,
and hence
P
has the fom
P
=f: 0cP
by means of
and PO'~)
A =
P01<)
<
00.
and
P.
If a sequence (P)
of stationary infinitely divisible distributions con-
n
verges weakly to
eral, 1. e.
Pn
~
Moreover,
uniformly in
then the representation
P,
is also infinitely divisible.
P
with respect to
P.
Ey,y- 1.....P
stationa;py infinitely divisible distPibution
A
EA,S with
Both
Then
00.
From this we see the validity of*
1......
y,y- P
4.5.
In this case therefore,
Lk>m
n.
4.3. is possible for P in genThe
all tend to the corresponding
nA
-+ 0
I;K;k
as
m -+
00
A0; ;x 's
with respect to
0
for every bounded interval
On account of the convergence of
I
to
this
relation is equivalent to
00
00
Um
I
n -+ 00 k=l
=
I
k=l
AI . K. k •
"
Both the above conditions are also sufficient for the convergence to
In other words,
P ".. .0> P if and only if for all
n
over for all bounded intervals
Um
n+
oo
Pn (X
Hr::H,
Pn (H)
-+
P(H)
P.
and more-
I
n (IxK) =f: cP)
= P(x
n (IxK) =f: cP).
Using known results on the distributions of sums of independent random
vectors, we obtain the result
*
Note added in proof: From equation (*) we have a further characterization
of the distributions EA S:4.5'. A stationary infinitely divisible distribution P has the form EA,S'if and only if P({cP}) is positive.
10
Let Pn, l' ••• 'Pn,s be a trianguZar aPray of stationary distributions
n
on MK satisfying aondition (2) of the definition of infinite divisibiUty.
Then the 1JJeak aonvergenae of (P 1 * ... * P
) to P as n+ 00 is equi4.6.
n,
vaZent to
(E
P
l , n,l
* ••• *E
A distribution
n
) -~ P.
P
l ' n,s
n,s
n
P is stationary and infinitely divisible if and only if
it can be represented as the limit of a weakly convergent sequence of distributions of the form
EA,S'
We have namely
4.7. For au' stationary infiniteZy divisibZe
1JJeakZy to P as
n +
P,
(mIn
P
aonverges
00.
We define here the fractional convolution power p
--J
EplIn
n,
m/n
by means of
= -nm"'"P.
5.
Regular and singular infinitely divisible distributions
We call a stationary infinitely divisible distribution
Zar infiniteZy divisibZe if for every
pew n «-t,t)xK)
¥
t
>
P on M
K singu-
0
~I~ n «(-t-m,-t) u (t,t+m»
x K) =~)
+
0 (m + (0).
This definition is intuitively clear albeit formally complicated.
We
shall now give it in a simple abstract form.
Let
P
~ o~
be stationary and infinitely divisible.
Consider the vectors
[N(wn«-t-m,-t)xK», N(wn«-t,t)xK», N(wn«t,t+m)xK»].
Let
r
denote the convolution of all distributions
eXp(-Ao;x)
I:=o
«Ao;x)m/m!)omx where
A. refers to the three intervals in the
11
vector just given and x
[m ,m ,m ] with nonl 2 3
runs through those vectors
negative integer-valued coordinates for which
m1
= m3 =
° and
m2 > 0.
Then
it follows that
I A•• X
x
~
P(x
=
n «(-t-m,-t)u(t,t+m»
is monotonically decreasing in m.
s.
Then
is precisely
\' A
!..X ';X
-+
p
5.1.
The conditional probability in the definition above
r(x'; [0,0,0]).
° as
Denote the convolution of the remaining
m=O
= r*s.
•
m -+
EA,S = 0$
m~
This tends to zero as
A stationary infiniteZy divisibZe distribution
EA,S
P(N(X) < /Xl)
S(M'K) = 1
all the mass of
P
EA,S
is concentrated on
P
For example, the convolution P
(n = 1,2, ••• )
P=
of E / P
l n, l/n
I:=l ~ Pl / n
nitely divisible.
lowing manner.
F(O+) = 0,
P
= EA,S*H
M'K'
necessarily have this form.
is singular in-
is no longer finite.
There are even singular infinitely divisible
be expressed in the form
on MK is singu-
S = 0$ we have
However, not all singular infinitely divisible
finitely divisible but
if and only if
= 0.
is singular infinitely divisible: for
and for
00
lIenee we have the following characterization.
00.
ZaP infiniteZy divisibZe if and onZy if
Every
= $, X n «-t,t) x K) .; $)
I
exp(-A .;y )
by
x K)
with
P which cannot in any way
S(M'K) = 1
and H singular infi-
Distributions of this type can be constructed in the fol-
Let
f~xdF(X)
F be a distribution function with the properties
= /Xl.
Then there exists no stationary renewal process
However, an analogous stationary measure
total mass can be defined.
QF on
[M'K'
QF satisfies conditions (8)
M'K]
and
P •
F
with infinite
(y).
All of its
12
mass lies on the measurable subset
0:_
x:
{
~I
t
t -+ 00
-1
N(Xn(O,t}}
= oJ.
A stationary infinitely divisible distribution
P
on
MK
is called
regular infinitely divisible if the conditional distribution
p(·I~ n «(-n-m,-n) u (n,n+m}) x K)
converges weakly to
P as m,n
-+
= ~)
00.
We shall now give an alternative form for this definition also.
We con-
sider the vectors
j,;.
[N(~n«-n-m,-n}xK}},
for fixed positive
t.
N(~n«-t,t}XK}},
N(~n«n,n+m}xK»]
The regular infinite divisibility of
P is then equi-
valent to the validity of
~'>.
Urn
m,n
-+
L
00
0
ox
m,n
=
for all
t > 0,
p(~n«(-n-m,-n)u(n,n+m})xK)
lim
=
'
-+
=
~,~n«-t,t)XK)
+ ~)
00
p(~n«-t,t)XK)
+ ~)
and we have
A stationary infiniteZy divisible distribution P on MK is reguZarinfinitely divisible if and onZy if P(N(X}=oo} = o.
5.2.
From 5.1 and 5.2, there follows
5.3.
Eaah stationary infiniteZy divisibZe distribution P on MK aan
be expressed in exaatly one way as the aonvolution Pl*P Z of a regularinfinitely divisible stationary PI with a singuZar infinitely divisible PZ'
13
We denote the measurable subset
X in M';{
Set
can be put in the form
MfiKnM =
K
M"K' Then [lfI,t]
+
of
{x: 0 < N(X) < oo}
{[zl,kl]"",[zn,kn ]}
T_tlfl
~
by
with
MilK'
Each
zl <••• < Zn •
induces a one-one measurable (both
ways) mapping of the direct product of
and
l
R
onto
MilKO
M*K = M*KnMK' Then to each measure on MilK there
Put
corresponds a well-defined measure on
M*KxBl where
B
l
is the a-algebra of
l
R •
Borel sets in
Now let Q be a distribution on M*K with the property
f:
and suppose
A > O.
also, for all
Q(w
Then by
H in
On the other hand,
~
n, A/n,Q
=
(H)
p
n, A/n,Q
where
4.7 E P
<
~ P',Q
00
as
n
+
and hence
00,
A
H,
(E
QXA~
# $)dx
n «I+x)xK)
nPA/n,Q(H)
nPA/n,Q
+
PA,Q
(n+ oo ).
tends to the measure induced on
denotes Lebesgue measure on
and it is clear that all
PA,Q(H)
Bl •
MilK by
Thus we have determined
are regular infinitely divisibleo
Now suppose that we are given a stationary measure
fying the conditions (S) and (y).
V on
MilK satis-
If we denote the corresponding measure on
M*KxB1 by VO, then for all S in M*K and bounded Borel sets B
product representation
measure on
M*K
Because of (y) ,
o
VO(sxB)
We have
VO
""
PA,Q
= G(S)J.J(B)
with
G(S) <
= GX~ = QXA~
with
A = G(H*K)
f~Q(Wn«I+x)xK)#$)dx <
00.
00.
Hence we obtain
~e have a
G is a finite
and
1
Q = >: G.
14
5.4.
A stationa:ry distPibution
P rf
divisibZe if and only if theX'e ewist
A >
°ep
on
MK is X'egula:P infinitely
0 and a distPibution Q on M* K
with the pX'opeX'ty
f: Q(~n«I+x)xK)
rf ep)dx
<
00
P = PA,Q' Both A and Q a:Pe unique ly detennined by P.
r
Let :H K and ~K denote the sets of infinite ~E~ for which
such that
N(~n«-oo,O)XK»
<
and
00
N(~n«O,oo)xK»<
respectively.
00
Both sets are in-
and are contained in M • If P(M r K) > 0 for
K
some stationary infinitely divisible P then we can proceed to a new distri-
variant with respect to T
bution S with
5.4.
5=
p«o)nMr K)
But these lead to
Q on
M'K
t
a
and apply the arguments used in the proof of
contradiction because the corresponding distribution
then cannot satisfy the condition
we have -P(M r K)=O.
Sim:Llarly -P(Ml K)=O.
f~Q(~n«I+x)xK)rfep)dx <
00.
Hence
These equations enable us to formu-
late the basic definitions of this chapter "one-sidedly":
5.5.
An in,fi11iteZy divisibZe Btationaroy cliBtX'ibution
on--Mk-u-{,B X'egu-
P
Za:P (Pesp. singuZaP) infinitely divisible if and onZy if
p(ol~n«-n-m,-n) x K)
(Pesp.~
fop aU
t
~
P as
m,n
The Poisson process distributions
P A = PA,o{O}
for
A> 0
x K) = ep)
+
0
00
as m +
00).
PA are all regular infinitely diviand
PA = 0ep
for
5.4 we then have for A > 0 the relations PA(N(X)rfl}
PA(xn(O,t)rfO} = At.
+
> 0,
P(~n«O,t) x K} rf epl~n«-m,O)
sible, for
= ep}
A = o.
= 0,
On account of
15
Hence a sequence
{p}
of stationary infinitely divisible distributions
n
°
PA if and only if for all t > 0, Pn(N(Xn(O,t»>l) ~
and
P (N(Xn(O,t»=l) ~ At as n ~ 00. With the aid of 4.6 we now obtain (cf. [25],
converges to
n
[28])
5.6.
Suppose given a trianguZa:r' a:r'l'ay Pn, 1"" ,Pn,s of stationa:r>y disn
tributions on M satisfying aondition (2) at the beginning of Chaptel' 4. Then
(P
n, 1*" .*Pn,s ) ==> P,A
n
if and onZy if fol' aZZ
s
t > 0,
n
L
lim
n ~
00
i=l
s
P i(N(Xn(O,t»
n'
> 1)
n
L
£lm
n ~
00
P i(N(Xn(O,t»= 1)
i=l n '
= 0,
=
At.
STATICI'.JARY STOCHASTIC POINT PROCESSES,
HI*
P. Froanken~ A. Liemant and K. Matthes
6.
(--J
A new approach to p;\,Q
We begin with a generalization of the construction of
PA,Q
given in
Chapter 4.
Let there be given a random mapping from k€K
to a cluster
~
with marks
,K'] ~ i.e. with each k€K there is associated a distribution Q(k) on
MK, such that for all C€MK, the real function Q(k) (C) is measurable with
respect to K. We shall assume that the marks of all marked points [x,k] of
in
[K'
generate clusters
~€~
Let
~
[x~k]
distributed independently according to
denote the measurable set of those
G
l
the sets
o[x,k]
By
joint.
~
Q(k)·
for which with probability one
of positions of elements of T-x~[ x, k]
G we denote the measurable set of those
Z
~
are pairwise disfor which, with
probability one, every bounded interval contains only finitely many positions
X~
of elements of
contains
~lu~2
=
if
tains all subsets of
For
p(~).
~€G
u[x,k]€~ T-x~[x,k].
~lU~Z€MK.
Set
G = GlnG Z•
If
~l'~Z
€ GZ'
~€G,
then
denote the distribution of the marked point process
Then for all
then
GZ
G also con-
~.
X~
by
X€M K, the real function p(o)(X) defined on G€MK is
measurable with respect to MK,
so we can associate with every measure
M satisfying the condition V(G) > 0
K
VQ( ) (X)
'*
Clearly if
=
a measure
V
Q( )
I P(~)(X) V(d~).
G
Jahro. Deutsch. Math.-Veroein.67 (1965) l83-20Z.
on M ,
K
V on
by means of
17
If Q{k)
does not depend on k we write simply VQ•
and it should be noted that
PA(G1)
=1
Thus PA,Q = (PA,)Q'
always.
From the definitions above, the following results are immediate.
(a)
~l€G
If the positions of the elements of
those of
are all different from
exists
~2t::G,
and P{~lU~2)
= P{~1)*P{~2)·
= V{~).
(b)
VQ{ ) (~i)
(c)
If V is stationary, so is
(d)
If every
c
i
>
V
Q( )
then the existence of
0,
to the existence of every
{Vi)Q
'
and
( )
From (a) we obtain
(e)
Let A, B be finite measures on MK for which A*B is meaningful.
Then (A*B)
= A
*B
in the sense that the existence of one
Q()
Q() Q{)
side implies the existence of the other and equality.
From (a), (b) and (e) we obtain
(f)
P
is a stationary infinitely divisible distribution if Pis.
Q( )
We retain the notation EA,S
tionary.
(g)
~co
-
L.m=O exp( -A)
for
Am
m
;r S
even if
S is not sta-
From (d) and (e) follows
If
EA,S
exists, then
(EA,S)Q()
= EA,SQ{)
in the sense that the
existence of one side implies the existence of the other.
The main result of this chapter is
6.1.
Then
(y)
Let P be a stational'Y infinitely divisible distPibution on MKo
e:x:ists if and only if (P) Q
e:x:ists and satisfies aondition
( )
( )
of 4.4 ° FoX' all X€MK, suah that cj>4x we have
PQ
---l
(p
Q( )
(X)
=
18
Proof:
We proceed in several steps.
(1 °> An analysis of its proof shows that the assertion
4.4 also holds
for non-stationary infinitely divisible distributions
,..,
sures
(8) and
V thereby satisfy requirements
V(wn({x}xK)~~) = O.
always that
(y) of
V.
The mea-
4.4. We have
Also in the non-stationary case
equation (*) holds as well as the correspondingly modified assertion
4.5.
(2°)
For all bounded intervals
ping of
into itself.
~
ping into a measure
for all
IV.
W+ wn(IxK)
I,
Every measure
IV
gives a measurable map-
V is carried by this map-
is infinitely divisible if
K such that ~¢X we have
V is, and
(~)(X) = I (V) (X).
XEM
Clearly
,..,
the measure
(3°)
IV
is always finite.
We now make the additional assumption that there exists
a > 0
with
the property
(k
Then every infinitely divisible
E
K).
V satisfies V(G 2) = V(G2) = O.
Moreover, it now follows from the existence of
measure also satisfies condition (y) of
Suppose
with
<
s.
= ip(~),
S
-l
A
r
Set
I = (r-a,s+a).
= A-l~.
I
equivalent to the existence of
From
(V)Q
that this
( )
4.4.
4.5, I P has the form EA,S
According to (g), the existence of
(~)Q
'
and
= E
(IP)Q
A'SQ( )
P
is equivalent to the existence of
Q( )
Hence the existence
The corresponding result holds for (P>Q
( )
( )
assumptions, the existence of
(IP)Q
( )
( )
statement of 6.1 is deduced.
From
it follows that
is
19
i.e. ,
=
We deduce from this that for all measurable
~¢X,
X with the property that
=
The remainder of the proposition follows from the fact that for all
with
[ml, ••• ,mn ]
~-.--~J
~
[0, ••• ,0],
E
K'
and
rt!
-1
f«IP)Q
Ll, ••• ,Ln
Il, ••• ,I n S (r,s),
)(H) =(PQ (H) and (IP)Q
(H) = (P)Q
(H) hold.
()
()
()
()
(4°) Let Q() be arbitrary. Then for every 1= (-n,n),
I(Q(»
satisfies the additional assumption used in (3°).
tational convenience here in (4°), we use
vals of the form
hence so does (P) Q
I ( )
Y
(n
~
= {4>: N(JlxL l )
[ml,.o.,mn ] ~ [0, ••• ,0],
00). Hence P Q
~ P
Q
I
given in II under 4.5, each (P
Let
I Q().
for all
(P) Q
I
IGI
Each
For no-
to denote only inter-
Then a fortiori every
~
m1, ••• ,N(Jn xLn )
~
P
I Q( )
exists and
mn }
P () (Y) increases monotonically to PQ() (Y)
IQ
as n ~ 00. From the continuity assertion
.LLJ
( )
I
=
(-n,n).
P
exists.
Q( )
For every
Assume now that
with
I Q()
both
_ ""
(----.J
{ ) (Y) - (P)I Q( ) (Y) converges to PQ() (Y).
IQ
be the analogue of Gl with respect to the cluster distribution
P(IGl)
increases monotonically to
P(G l ).
Since P{IGl) = 0
n,
P(G ) = 0 also. Now if P(G2) > 0, we cannot have
l
(N(4>n(JxK'» ~ m) converging uniformly in I to zero as
( )
Hence the existence of
(P)Q
is established.
( )
m ~ 00.
20
Y,
For all
I
Therefore
(P)Q
increases monotonically to
(Y).
( )
(X)
X€M K,
for all
( )
(P)Q
(P)Q
( )
= PQ
(X)
"'( )
particular,
(Y)
({P) Q
satisfies condition (y) of
with the property
$€X.
In
4.4.
( )
(5°)
Conversely, we now assume that
dition(y) of
(P)Q
exists and satisfies con-
( )
4.4. Then the same is true for
Hence
Q
(P)
I
( )
P
exists also. Let V be the infinitely divisible distriIQ( )
bution corresponding to (P)Q
« )n~~$). For all Y (P) Q (Y)
increase monotonically to
I
'" ( )
(P)Q
(Y)
as
n
+~.
( )
By the continuity
( )
assertion already used in (4°) we see that
to
V as
n
P
converges weakly
I Q( )
P
follows from the same
Q( )
The second
results used in (4°) to show the existence of (P)Q
+~.
The existence of
( )
statement of 6.1 has already been proved in (4°).
Thus the proof of
6.1 is complete.
In general,
(P)Q
({$}) > 0,
so the equation in 6.1 cannot hold for all
( )
measurable
X.
The restriction
$~X
is superfluous if
Q(k)(P~$)
= 1
for all
k€K.
From 6.1 and the easily determined structure of '"
P
-.-J
special case the structural assertion for (PA,Q
A
there follows as a
derived towards the end of
Chapter 5.
Also from 6.1, we obtain the following sharpening of the second invariance
property of P
6.2.
A
given in Chapter 3:
Let P be the distribution of an independentZy maPked stationcwy
Poisson pl'oaess.
If Q(k) (N(P)=l) = 1 fol' aU k€K,
then P
Q( )
exists and
is aZso the distnbution of an independentZy maPked stationcwy Poisson pl'Oaess.
We have already used 6.2 after 4.1 to prove that
PA,Q
= PA,Q*'
21
7. Palm distributions in a generalized sense
Let
SK
denote the class of those measures on M
K
stationary with
respect to T and satisfying condition (y) of 4.4. Although any Q~SK
t
need not be finite, a substantial part of the remarks in Chapter 2 remain valid
for such measures.
In contrast to Chapter 2, we will not distinguish here any LE:K,
develop ideas in this chapter only for the special case L=K.
Le. we
Accordingly, the
intensity PQ of QE:SK will always be understood to be the integral
fN(~n«O,I)XK»Q{d~). For
distribution of
Q,
0 < P
Q
<~,
we can then introduce
QO'
the PaUn
by
=
Jl
fN({X: O<x<l, ({x}xK)n~~~, T ~E:S})Q(d~)
P
x
Q
SE:MK• Q is always a distribution on MK, and in the case where
O
Q(M'K) = 1, Q({~}) = 0, it coincides with the Palm distribution (with respect
for
to
K) introduced in Chapter 2.
Set MO
Q
O
= {~: ~E:MK,({O}xK)n~~~},
MO K = MOKnMK• Every Palm distribution
K
can b~ regarded as a distribution on
MO K•
The foliowing formulae for computations clearly hold.
7.1.
Let Q = liQi with Q,
QiE:S K,
QO = pI liPQ (Qi)O'
Q
i
7.3.
By 7.2
Fop aZl Q€SK such that 0
Q <~,
< P
(Q«
)n~~~»O
= QO'
P
can take any positive value for a given Palm distribution QO.
Q
22
WK be the set of all Q€SK such that Q({~}) = O. From the results
of Chapter I of [30], we then have the following inversion formula:
Let
7.4.
Fop aZZ
Q€WK UJith finite positive
non-empty bounded open intepvaZs I = (a,S),
Q(S n
UJhepe
x
o(~)
Each
(~n(IxK);t~»
=
P
Q
f[max(a.S-xO(~»
t
~
= 00 UJhen no e Zement in
Q€WK such that
0 < P
Q
intensity~
<
00
aZZ S€MK and aZZ
ks(T_TtldJ
J
has its position in
QO(d~l
(0,00) •
is thus uniquely determined by
P
Q
>,,-
and QO.
We denote by
ing 4.5.
(n
~
~
the type of convergence in WK introduced in II follow-
If Q as well as all the
is equivalent to Q
(0)
n
=l>
Qn's are distributions, then Qn
Q (n
The transformation from Q to
~
~
Q
(0).
Q is continuous in the following sense
O
(cf. [29]).
7. 5.
P
~
Qn
(n
~
P
Q
s
(n
~
(0).
Q,
Then
QncWK UJith
Qn-~
Q (n
0 < PQ,
~ (0)
P
<
Qn
00
(n
= 1,2, ••• ),
is equivaZent to
and
(Qn)O =<> QO
(0).
Let
Q€W
Suppose
s
W K denote the set of all
Q€WK such that Q(~fM'K) = O. For all
the inversion formula1 2.7 is valid. In [30], the
with 0 < P < 00
K
Q
following generalization of 2.9 is proved:
1
The proof of 2.7 given in I contains a gap: both sides of (5) there may be
infinite. In the proof of 2.7 however, we may restrict attention to such S
for which ~€S always implies YO(~)-y-l(~) ~ Cs with Cs < 00. Then
J[J~O kS(TT~)dT]P(d~) ~ c .
s
23
7.6.
1J)ith 0
s
A distPibution P on MD K is the Palm distrribz.i.tion of some Q€w K
< P
Q
O
<
co if and only if P(M' K) = 1 and the !'estPiction of P to
is invaPiant mth !'espeat to
M K OM ' K
0.
if
zn
= o.
Then lITe have
7.7.
r
A distPibution P on MDK is the Palm distPibution of some QE:W K
mth 0 < P < co if and only if p(lP is finite) = 1 and the !'estPiation of
Q
P
r
to w K is invaPiant
7J)i th
respect to y.
7.6 and 7.7,give us a general view of the class of all Palm distributions
of measures in W since, as we saw in Chapter 5, every QE:W is uniquely
K
K
r
expressible in the form Q = Ql +Q 2 with QlE:W K , Q2 €ll K.
We now introduce a simple class of Palm distributions.
Let
F be a left-
continuous distribution function of a positive (possibly infinite) random variable.
ThuB we require
F(O+O)
=0
but allow
F(co) < 1.
Let
(~i)-CO<i<co
be a doubly infinite sequence of independent positive random variables with
distribution
F.
If
F(co) = 1
then with probability 1 all
so that we can form the point process (as under 2.9 in I)
~
i
's
are finite
24
However, if
F(oo) < 1,
negative index n
such that
~
n
=
Z
Thus with every
such that
l
then
~
n
=-
and a greatest negative index n Z
00,
l
We then consider the point process
00.
F we have associated a distribution
of 7.6 and 7.7 every
F(oo) < 1
then with probability one there exists a least non-
H
F
Qe:ll;
HF on MO.
is a Palm distribution of a measure
otherwise
Qe:~ •
W.
Q on
<
00.
In this case, we can set
f~xdF(X)
=
00,
Q is an element of the family2 of measures
Q
= PF •
For
F(oo)
=1
and
QF given in II
5.1.
8. The distributions
(P)o
From now on we again assume that condition (a) of Chapter 4 holds.
by 4.4 we have an invertible one-one mapping of
stationary infinitely divisible distributions on
s
(W )
K
If
Q is finite if and only if
f~xdF(X)
under
On account
WK onto the class
MK•
Then
UK
Here the classes
correspond to the regular (resp. singular) infinitely divisible
of
r
WK
P
in
UK·
By simple computation, we have (cf. [Z9])
8.1.
Pp
= Pp
~henever
Thus in the case
of 4.4 and 7.4 each
2
Note that
multiplier.
0 < Pp <
0 < Pp <
Pe:U
K
00,
00.
we can form
(P)o as well as PO. Because
with finite positive intensity is uniquely
QF was uniquely determined only up to a positive constant
25
determined by
8.2.
Pp
+
and
(P>O'
For a given
(P)O'
Pp
From 7.5 and the remarks in II following 4. 5,
value.
Pp
Pp
n
For
'toTe
ob tain
Let P, Pn€UK with 0 < Pp ' Pp < w (n = 1,2, ••• ) and suppose
n
(n + 00). Then Pn ~ p is equivalent to (Pn)o ~ (P)o'
= E>. ,S
P
A > 0,
with
....,
PP = AP S '
can assume any positive
(P)
0
0 < P
= SO'
Po = p*(P)O·
Thus 1f
=1
F(oo)
and f~xdF(X)
< l?OJ
then the class of those
P
of
On the other hand, we do not know any simple repre-
E p
with
A, F
A > O.
in
we find by computation that
< 00,
s
....,
U such that
(P)O = HF is precisely the set
sentation of the singular infinitely divisible
for
F(oo) = 1,
For all
n nip +
P
f~XdF(X) <
K with finite positive intensity we have by 4.7 the relation
and hence using 7.5
'toTe
P be any distribution in
(En~n/p)o = (En,n/p)*(n/p)o'
8.4.
Then by
(P)O.
have the convergence
UK with
~ Po
0 < Pp < 00.
as
n
+
00.
Then because of
However,
and here the first factor converges to
P
and
Using 1.3 we obtain
For all P€UK suah that
Now let
(P)O = H
F
00.
4.7 and 8.2 we have the relation (En,n/p)o
the second to
such that
P€U
(n + 00),
Now let
P€U
0
<
Pp
<
00,
P€UK be regular infinitely divisible and distinct from
5.4, p = P>.,Q with >.
> 0
aud
Q(M*K) = 1,
o~.
whence
Pp = P...., = AfN(~)Q(d~).
P
For the time being, we make the additional assumption that
Let
r
be a random variable with
p(r-i) = lin
(i = l, ••• ,n)
Q(N(~)=n)
= 1.
independent of
26
the marked point process
~
~
with distribution
Q.
With probability one each
is of the form
o = zl
with
Q+ denote the distribution of
Now let
particular, for
A > 0,
'"
(PA)O
(~)~.
Tz
r
is precisely
We now turn to the weaker restriction
PA,Q = In
8.5.
~hepe
< ••• < Z •
n
In
Then
o{O}.
0 < Pp <
00.
Then
Q(~=n)PA,Q( I~=n) and hence
«(~)O
=
(fN(~)Q(d~»-l
In
nQ(N(~)=n)(Q( IN(~)=n»+,
the summation is taken ovep aZZ n suah that
Q(N(~)=n) >
o.
'"
(P)O
= H for F(oo) < 1 is
F
regular infinitely divisible, and accordingly all such P are of the form
We already know that any
PA,Q.
Clearly
P€U· such that
Q is uniquely determined by
On the other hand
F
if we require
A may take any positive value.
Q(M*) = 1.
From 8.5 the form of
Q
is easily obtained:
8.6.
Equation
8.4 is easily interpreted for P regular infinitely divisible.
We see that it can even be used as a characterization of infinite divisibility,
viz. (cf. [30]) we have
on MK ~ith 0 < Pp < 00
thepe exists a distPibution H on MO K suah that Po = P*H, then P€UK and
H = (P)o. In paPtiauZaP a stationary distPibution P on M suah that
8.7.
o < Pp <
If fop a stationary distPibution
00
P
is of the fo1'T'fl P A if and onZy if Po = P*o{O}'
27
The particular case in 8.7 has already been given by Slivnyak [18].
9.
(1 s i s S n ; n = 1,2, ••• ) of stationary distributions
is called infinitesimaZ if it satisfies condition (2) at the beginning
array
An
M
K
on
Generalization of a result of Palm and Khi-nchin
(P
of Chapter 4.
n,
i)
From 4.6 and 8.2 we have
9.1. Let
(P
n,
be an infinitesimaZ array of stationary distributions
i)
on MK with finite positive intensities P i ' If there exists a distris
n,
bution P suah that 0 < Pp < co; Li:l Pn,i ~ Pp s (n -+ co) an~
(Pn, 1*'" *pn,s ) ~ P
(n -+ co),
n
~ (P)O'
then P€uK and
n, l*"'*Pn,B n ) ~ P,
it does not necessarily follow that
Given
(P
o
<
Pp
<
P = Pp '
00,
~ n
-1 ~ n
(L·1= lP n, i) (Li =lP n, i{Pn, i)O)
and
s
Li:l Pn,i -+ P as
For example, if
(Pn,i)
n -+
00,
satis-
fies all the assumptions of 9.1, then the array formed by adding Pn,s +1
=
n
(A > 0)
E -1
n
'PnA
(n -+ (0).
is also infinitesimal and (P l*"'*P
+1) > p
s +1
n,
n,sn
However, Li : l Pn,i -+ Pp+A (n -+ (0). In general, we may conclude
from 0 < Pn, i <
Pp
S
00
and
(P n, l*"'*Pn,B )
~
P only that
n
B
Urn .innn+oo Li:l
- P
n,i ' However, under the assumptions of 9.1,
(Pn, i)
is
infinitesimal in a stronger sense:
If
9.2.
max!SiSs
n
(P n, i)
Pn,i -+ 0
Proof.
satisfies all the assumptions of 9.1, then
(n-+
oo ) .
Suppose that the maximum
generality, we can assume that
Pn, 1
~
+> 0
(n
c > 0
for all n.
-+ (0).
Then without loss of
Then since
-e
28
(P n,2 *... *Pn,s ) =>P,
<
Pp -
D:_
-vLITI
s
'1
\' n
-<.nun+<x> Li=l Pn,i ~ pp- c ,
which is a con-
n
tradiction.
A sharper converse of 9.1 is
9.3.
Let
(P
n,
i)
be an infinitesimal,
s
on MK with finite positive intensities.
(n
where
+ w)
0
Then
(n + w).
<
(P
of stationary distributions
aPt'ay
sn
*... *pn,s
-1
(Li=lPn~i)
P < w and that
n,l
Suppose that Li:l Pn,i
P
sn
(Li=lPn,i(Pn,i)O) ~ X
aonverges weakly to some
)
+
with intensity
P
n
The proof follows from 4.6, 8.2 and the following lemma.
9.4.
The alass of aU Palm distributions of measures on WK with finite
positive intensities is alosed under weak aonvergenae.
Proof.
Let a sequence
P
n
of Palm distributions converge weakly to a
distribution
X.
intensi ty 1.
Because of 8.2, it is sufficient to show that
some
P we associate the corresponding
n
With each
Qn€WK with
Qn
converges to
K with intensity 1.
Q€W
By a formal application of the inversion formula
in the a-ring
kR
defined following 4.4
kQ(S)
=
J[
Jk
ks(T_TO>d:J X(d.>,
J
max(-k,k-xO(<P»
thus obtaining a finite measure on
For
I , ••• ,I1
1
7.4 we define for all S
~
(-k,k),
kR.
1 € K,
Lp ••. ,L
and
[m , ••• ,m J
1
1
r:f:
[O, ••• ,OJ,
let
Then the bounded integrand in square brackets. is in a certain defined sense (see
29
[29]) continuous almost everywhere with respect to X.
the form specified,
Qn(S)
~
kQ(S)
From this it follows that
(n
~
Thus, for all
MK
Q({~}) =
such that
k+lQ is always an extension of kQ•
satisfies condition (y) of
O.
Therefore
kQ to a measure
On the basis of the definition of
kQ,
4.4.
n
~ 00).
Therefore
Q
Q also
The convergence statement given above can now be put in the form Q
(n
of
00).
since (a) holds, there exists a common extension of every
on
S
Q is also stationary and hence in WK'
~
Q
Clearly
Now
lim
t .j.
0
~ t~O ~ f[t-xDm/j X(d~)
~ Q(~n«O,t)xK) # ~)
If we had used here the representation
would have been
PQ.l.
Hence
P
Q
= 1,
7.4 for Q, then the right-hand term
and the proof of
Clearly a sequence of Palm distributions
converges to
(PA)O = o{O}
(A
>
0)
l.
•
Y
n
9.4 is complete.
of distributions in
if and only if for all
t
>
0
W
the se-
F
of xO(~) induced by Y converges to
n
n
If we note that to a mixture of Palm distributions there is a corre-
quence of distribution functions
zero.
sponding mixture of distribution functions of
xO(~)'
then we obtain from
9.1 and 9.3 the sharper form derived in [23] of the result of Palm and
Khinchin on convergence to
9.5.
Let
(P
n,
i)
P
A
(prov~din
be an infinitesimaZ arTay of 8tationarry distPibutions
s
on M with finite positive intensities
(n
~ 00),
0 < A < 00.
equivaZent to
[4]):
Suppose
Pn,i'
Then the convepgence
(P
n,
l*"'*P
Li:lPn,i ~ A
n,s
)
n
~
P,
1\
is
30
Sn
l
, 1
J.=
where
Fn, i
p
n,i
F
n,
i{t)
+
for au'
0
is the dis tribution function of
X
> 0,
t
o(~)
induaed by
(Pn ,i) 0
(cf. 2.7 of!).
10. limit distributions for infinitesimal arrays of stationary
renewal processes
G denote the class of all left-continuous distribution functions G
Let
of non-negative (possibly infinite) random variables.
always, but it is not necessary that either
G
n
=i>
G
=0
G{O+O)
if for all continuity points
(n + 00)
t
We have
of
G,
or
G{O)
G{oo)
=0
= 1.
Write
G (t) + G{t)
n
(n + 00).
F denote the class of all
Let
FEG
such that
F{O+O)
7, we established a one-one correspondence between each
tributions
H on
F
= O.
In Chapter
FEF and certain dis-
MO. The convergence just defined for distribution func-
tions corresponds to weak convergence of the corresponding distributions on
10.1.
only if
F
n
For a sequence of distributions
==;>
F
(n + 00)
for some
FEF
FnEF,
and also
H
=l> X
F
n
X =H •
F
(n -+ 00)
if and
The proof follows from the simple lemma 10.2 below.
~EMo
Each
Let
has the form (cf.
denote the "lJ.'fetJ.'mes"
termined by
(dj ) -m- l<J<n'
l'
1.1 of I) {xi}-m-l<f<n' where X_I
x j+l-x j'
Then
Write the event
meMO
~'"
= O.
is also uniquely de-
31
-m-l < r < s < n-l t
d < t
i
i
for
r SiS s
more briefly as
{di < t i t rSiSs}. Then from the characterization of weak convergence derived in [29] we have the lemma
10.2.
A sequence
of distributions on MO
(Y)
n
if and onZy if for' aZZ
such that
r < s
r
Y
and aZZ continuity points
ass
S
conver'ges 1JJeakZy to
With the aid of 9.1 t 9.3 and 10.1 we may stmlIIlarize the limiting behaviour
of infinitesimal arrays
(Pn,i)
with constant "rows"
(Pnti)ISiSS
of stan
tionary renewal distributions.
of I that
(pp )-1
F
10.3.
fJ.
<
F
Let
fJ.
00 t
n
F
-+
= fJ. F = f~
(F)
n
00
We note here the relation derived under 2.9
xdF(x).
be a sequenae of eZements of F such that Fn (OG) = 1,
(n -+ 00).
Zet
FU:r'ther'~
n
be a given sequence of non-
(8)
n
negative integer's such that Sn(fJ. )-1 = SnPF -+ P (n -+ OO)t a < P < 00. Then
Fn
S
n
(P ) n d> P (n -+ 00) is equivaZent to F ~ F (n -+ oo)t F€F, Pp = p,
F
n
n
and
""
(P)o
= HF •
The
F appearing in 10.3 may be any element of
10.4.
Fn
~
F
To
(n
-+
evePy
oo)t
F€F the:r-e e:x:ists a sequence
Fn(OO)
= 1,
fJ.F <
00,
Proof. Suppose F€F and F(ta)
F (t)
n
where
fJ.
n
t
n
__
{(II -
= 1.
;> F(t)
F
n
-+
F€F has
F(t) < 1
(F)
n
in F such that
00.
Set
for
t S tn t
for
t > tn'
is chosen sufficiently large so that
on the other hand t
F on account of
for all
fJ.
F
n
<
o
> n
t
<
(e.g.
00,
t
n
= n 2 ).
then set
If,
32
=
F (t)
n
where
t
{
F(t)
t
F(n)
n < t S t
S n
is chosen sufficiently large so that
n
n
1
6
> n
F
(and hence
n
This choice of
suffices to prove the assertion of
F
n
10.4.
The examination of arbitrary infinitesimal arrays of stationary renewal
distributions
P
requires some preliminaries.
F
n,i
First we recall the concept of weak convergence of distributions (cf.
[3lJ, [32J).
E be a complete separable metric space and E the a-field
Let
E.
of Borel sets of
A sequence
weakly to a distribution
tinuous real functions
topology in
P (L)
n
~
E.
P(L)
P
f
From P
n
n
P
of distributions on
!fdP ~ !fdP
if
on
~
(P)
n
E.
(n ~ ro)
E converges
for all bounded con-
Thus weak convergence depends only on the
(n
~
ro)
follows the "pointwise" convergence
for all continuity sets relative to
P,
i.e., for those sets
LEE whose boundaries have zero probability relative to
troduce on the class of all distributions on
P.
We can always in-
E a metric with respect to which
this becomes a complete separable metric space in which the metric convergence
coincides with the weak convergence introduced above.
For
E
=
[O,roJ
the distributions on
with the distributions in
vergence
G =<> G
n
erties given above.
We denote by
(n
~
ro).
E are in one-one correspondence
G whereby weak convergence corresponds to the conHence there exists in G a metric with the prop-
Because of the compactness of
G the family of Borel sets of G.
E,
G is also compact.
It is generated by the sets
{G: G(t)<c}.
Clearly all of
HF{di<t i , rSiSs}
measurably (with respect to
G)
on
and hence all
F,
~(S)
(SEM o )
depend
so that tve can form the mixture
33
fFHF( )q(dF)
for all distributions
q
on G such that
= 1.
q(F)
Then we
have
10.5.
q
f F HF (
is unique Zy detemined by
) q(dF).
Proof. Let h be a function on G of the form h(G)
Here we may choose for
(tv)
= nn
v=O
G(t).
v
an arbitrary finite sequence of positive numbers.
Then
=
IF h(F) q(dF)
The expected value of
IF
~{di_Z <
ti
h with respect to
for
q
1 SiS n} q(dF).
can therefore be expressed by
We now consider functions
is once
more an arbitrary finite sequence of positive numbers.
Each
form limit of a sequence of linear combinations of functions
f F f(F)q(dF)
f
is the uni-
h.
Hence
f F ~q(dF).
is also uniquely determined by
The same is also true for all finite linear combinations
g
= aO+L~=lakfk'
The class of all such
of all continuous real functions on G.
points in G,
~
g(G )
l
g(G )'
Z
i.e.,
G
l
~
g
forms a sub-algebra of the algebra
Clearly this sub-algebra separates
G
Z implies the existence of some g such that
By the Stone-Weierstrass theorem (see e.g. [33J p. 47), every
continuous real function on G is the uniform limit of a sequence
Thus we see that the integral with respect to
function on G is uniquely determined by
q
fFHFq(dF).
(g ).
n
of each continuous real
Hence our proposition
10.5 follows.
The transformation from
sense:
q
to
f FHFq (dF)
is continuous in the following
34
10.6.
=1
be a sequenae of distPibutions on G suah that
(q)
n
IF HF qn(dF) ~ X (n
X = IF HFq{dF).
= 1,2, ••• ). Then
(n
~ q
Let
(n
+
00),
q{F) = 1,
Proof. Suppose qn
~
q,
Ii = (ai,b i )
open intervals
q(F) = 1.
(i = l, ••• ,m)
=0
bl, ••• ,bm}~$)
H {N(9nI )=n , lSiSm)
F
i
i
Theorem 1. 8)
such that
Now suppose that
IF HFqn(dF) ~>
values
x,
Then, relative to
for almost all
~
q
n
q,
(n
(F)
is equivaZent to qn
00)
q,
Hence by 10.1,
F.
is continuous q-almost everywhere.
IF HF~(dF) ~ IF HFq(dF)
i.e.,
n
Choose a finite sequence of bounded
IF HF (9n{a l ,···,am, bl,···,bm}~$) q(dF) = O.
HF(~n{al, ••• ,am'
+
q
Hence (cf. [32]
00).
+
q(F) < 1.
Then it is impossible that
for otherwise by 10.2, apart from a countable set of
IF~(d_l<t)qn(dF) + X{d_l<t),
i.e.,
IFF{t)qn(dF)
right-hand side converges monotonically to zero as
t
~
+
O.
X(d_l<t).
t
The
On the other hand,
however,
t
~
~
tim
0 n
Now suppose
+
I
00 F
~
F(t) q (dF)
n
IFHFqn(dF) ~ X (n
+
IF F(O+O) q{dF)
00).
qn => q
(m
+ 00).
Then
o.
By the compactness ofG,
is relatively compact in the set of all distributions on G.
vergent subsequence
>
I F~qn
m
Choose any con-
(dF) ~ X (m
and
q(F)
By 10.5, all convergent subsequences of relatively compact sequences
have the same limit, i.e.,
+ 00).
m
By the first step in the proof, we have X = IFHFq(dF)
q
n
=> q
the proof of 10.6.
From 9.1,9.3 and 10.6 we have
(n +
00)
and
q (F) = 1.
(qn)
= 1.
(qn)
This completes
35
10.7.
Let
(P i)
n, s
2i : l
t:roibutions suah that
=;'>
(n +~)
P
be an infinitesimal, a:I'xoay of stationary reneuJal, disPn,i
+
P
(0
<
(P F *.. '*PF
)
n,l
n,sn
P < ~)., Then
is equival,ent to
sn
]-1 [ sn
J ~>
2
p
2
P
0
[ i=l n,i
i=l n,i {Fn,i}
q,
= 1,
q(F)
Pp
= p,
We end this chapter with
10.8. In 10.7,
aan be any distnbution on G suah that q(F) = 1.
q
Proof. In the space of all distributions on G fix a metric with the properties indicated earlier.
on,' G 1o1ith q (F) = 1 be given.
Moreover, let q
We now cons truct a corresponding array
(PF
by rows.
•)
n,1.
Let
n
be any non-negative integer.
2~=laio{F}' FiEF,
whose distance to
a mixture
First we choose
q
is less than
lin.
Here the coef-
i
ficients can always be chosen to be of the form
non-negative integers and k > n.
The proof of
be replaced by distribution functions
J~xdLi(X) = kr
m rri
2 =1 rk o{L.}
i
1.
for a non-negative integer
to
is still less than
q
row of the desired array as follows:
P
n,i
's
equal to
LiEF
PL'
As
i
n
a
= r/k where the r i are
i
10.4 shows that the F 's may
i
such that
r,
lin.
Li(~)
= 1,
and the distance from
We can now construct the n
let s n = 2mi_lr.r
- 1.,
an d set
increases, the mixture converges to
rir
0
f th e
q.
11. Some relations between P and PL
In what follows
P
Moreover, we assume that
throughout.
is a distribution on
P(~LEM')
= 1.
MK, L€K and 0
<
A(L)
< ~.
We use the notation of Chapter 2
th
36
First we start from the footnote following 2.3.
Let
S be any set in
ML • Then with probability one, the limit nS of
t- l N({x: O<x<t, ({x}xL)n~~$, T ~€S})
x
as
t -+
00
exists.
a distribution P*,L
We set
n;. = a
L
on M •
coincide if and only if a
and obtain by virtue of
PL , P*,L are equivalent.
almost everywhere. Thus, if
The measures
equals
PL = P*,Lo
From the definition of
A(L)
They
P is
ergodic,
we have
From the Ergodic Theorem follows the existence for almost all
relative to
PL of the limit
00
of
I rn-l
n L.i=O(Yi(~)-Yi-I(~»
as
dP* L
= (A (L) )00.
dP
L
Proof. By 2.7
11.2.
~,...z.2=
f(Yo(~)-y-I(~»ks(TYi(~)~)PL(d~)
=
A(L)
=
f A(L)(Y_i(~)-y_i_I(~»kS(~)PL(d~)
and hence
from which the assartion 11.1 follows by the ergodic theorem.
~€ML
n -+- co.
37
From 11.2 we see (cf. [18], Theorem 5) that
For mixing P we can deduce a simpler assertion.
by
$t
the restriction of T_
to
t
MK;
For brevity, we denote
then for mixing P we have the con-
vergence result
11.4. PLI$t
~ P
(t ~ ~).
Proof. Let Il, ••• ,It be bounded open intervals and Vl,.,.,Vl sets in
K.
Set
By 2.4, to every
for
If
0 <
T
€
> 0
there exists
T
€
> 0
such that uniformly in
t
and
T S T ,
€
is chosen sufficiently small then for
On the other hand, for large
t
sufficiently large
t
We can also sharpen 11. 1 under suitable assumptions:
11.5.
as
n
+ ~
If PL is mi::cing with respect to
to P (S)
L
fop atz
L
se:M.
The proof follows from the equation
0"
then P(T
Yn
(\p) \PES)
converges
38
The following example shows that the conclusion of 11.5 does not hold for
all mixing
Let
P.
Al
and
a
be positive and
(~i)-oo<i<OO
a dOUbly-infinite sequence
of independent positive random variables distributed according to
l-exp(-Ax)
(x
~
0).
Let
F(x)
=
X denote the distribution of
and Y the distribution of
By 2.9 the mixture
M'.
~(X+Y)
It is clear that
is the Palm distribution of a distribution
P
on
P is mixing, but the conclusion of 11.5 does not hold.
REFERENCES to Parts II and III
[18]
I.M. SLIVNYAK. Some properties of stationary flows of homogeneous random events. Russian. Teop. Vepoyatnost. i PPimenen. 7 (1962)
347-352. (Transl.: TheoPy PPOb. AppZ. 7 (1962) 336-341.)
[23]
G.A. OSOSKOV.
A limit theorem for flows of similar events. Russian.
(Transl.:
Teop. Vepoyatnost. i PPimenen. 1 (1956) 274-282.
TheopY PPOb. AppZ. 1 (1956) 248-255.)
[25]
B. GRIGELIONIS. On the convergence of sums of random step processes to
the Poisson process. Russian. Teop. Vepoyatnost. i PPimenen. 8
(1963) 189-194. (Transl.: TheopY PPOb. AppZ. 8 (1963) 177-182.)
[26)
P.R. HALMOS.
[27]
K. MATTHES. Unbeschrankt teilbare Verteilungsgesetze stationarer
zufalliger Punktfolgen. Wiss. Z. Hochsah. EZektPO. IZmenau 9
(1963) 235-238.
[28]
P. FRANKEN. Approximation durch Poissonsche Prozesse.
26 (1963) 101-114.
Measupe TheopY.
New York: Van Nostrand (1950).
Math. Naahp.
39
[29]
J. KERSTAN and K. MATTijES. A generalization of the Palm-Khinchin
theorem. Russian. Ukrain. Mat. Zh. 17 no.4 (1965) 29-36.
(Trans1.: Statistics Department (lAS), Aust. Nat. Univ.)
[30]
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