IThis research was supported in part hy the Office of Naval Research,
Contract N00014-67-A-0321-0002, Task NR042214.
2
On leave from Chalmers University of Technology and the University of
Goteborg, Sweden.
-,
ON SYMMETRICALLY DISTRIBUTED RANDOM MEASURES
Olav Kallenberg
l
2
Department of statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 895
October, 1973
ON SYMMETRICALLY DISTRIBUTED RANDOM MEASURES I
by
Olav Kallenberg
ABSTRACT. A random measure
~
defined on some measurable space (S,S) is
said to be symmetrically distributed with respect to some fixed measure
on
S, if the distribution of
only depends on (wAl, ... ,wA ).
k
(~Al, ... ,~Ak)
for kEN
w
and disjoint AI" .. ,AkES
The first purpose of the present paper is to
extend to such random measures (and then even improve) the results on convergence in distribution and almost surely, previously given for random processes
on the line with interchangeable increments, and further to give a new proof
of the basic canonical representation.
The second purpose is to extend a well-
known theorem of Slivnyak by proving that the symmetrically distributed random
measures may be characterized by a simple invariance property of the corresponding Palm distributions.
IResearch sponsored in part by the Office of Naval Research under Contract
N00014-67A-0321-0002, Task NR042214 with the University of North Carolina.
AMS (MOS) subject classifications (1970).
60FlS.
Primary 60K99; secondary 60FOS,
Key words and phrases. Random measures and point processes, symmetric
distributions, interchangeable increments, weak and strong convergence, invariance of Palm distributions.
Unclassified
Se-curity elllSSifiel' lion
,.,
..
""••,,, .....a".,,, • •, au., .., •• ~ ~~~~~.~.~~,-~?~.~:'~~.:,~ ~,?
.....,.. "......n...............,."
''''0''''11 "ulho,)
Department of Statistics
University of North Carolina
Chapel Hill, NC 27514
. ORIGIN .. TING "CT'VITV (I
a". JIl.POJllT II[CU"'TY C ..... II .. ,C .. TION
lin r 1 $I c;. c;. ; f' ;
A£1
Ill, GJIlOU'"
3 ... EPOJllT TI TL.IE
On SYmmetrically Distributed Random Measures
4. DESCRIPTIVE NOTES (Type of ,epo" end Inc/usiv£' dele.o)
Technical Report, October, 1973
s· "u THORISI (FI,st n"me. middle Inll/"/, le.' n"."e)
Olav Kallenberg
Il.REPORTD .. TE
acoer,
t b
1973
"". CONTRACT OR GRANT NO.
b.
NOOO14-67A-0321-0002
PROJECT NO.
Task NR042214
c.
711.
TOTAL NO. OF PAGES
22
r
b
.
N~3 OF
REFS
lIlI. ORIGIN .. TOJll·S REPORT NUMBERIS)
Institute of Statistics Mimeo Series
No. 895.
lib. OTHEJIl JIlEPORT NOIS) (Any olhe, numbe,. "',,' m"y be " •• ,,,..d
Ihl. repo,')
d.
10. DISTRIBUTION STATEME ~T
"
Approved for public release:
11. SuPPL.EMENTARY NOTE::;
distribution unlimited.
12. SPONSORING MILITARY ACTIVITY
Office of Naval Research
13. "BSTR"CT
A random measure s defined on some measurable space (S,S) is said to be symmetrically distributed with respect to some fixed measure w on S, if the distribution of (sAl, ..• ,sA k) for kEN and disjoint Al, ... ,AkES only depends on
(wAl, ... ,wA ). Theflrst purpose of the present paper is to extend to such random
k then even improve) the results on convergence in distribution and
measures (and
almost surely, previously given for random processes on the line with interchangeable increments, and further to give a new proof of the basic canonical representation. The second purpose is to extend a well-known theorem of Slivnyak by
proving that the sYmmetrically distributed random measures may be characterized
by a simple invariance property of the corresponding Palm distributions.
KEY WORDS:
Random measures and point processes, sYmmetric distributions,
interchangeable increments, weak and strong convergence, invariance
of Palm distributions.
Unclassified
1.
Introduction.
Let
8 be a locally compaCt second countable Hausdorff
B be the ring of bounded Borel sets in
space and let
Write
8.
M(s)
for
the space of Radon measures on (8,B), endowed with the vague or weak topology
[7], and let
be the sUb-space of
N(8)
Z -valued measures.
~
OOEM(s), we say that a random measure or point process
element in
M(S)
with respect to
or
00
N(S)
Given any fixed
+
S
on
(i.e. a random
respectively [2,5]) is symmetricaZZy distributed:
[6], if for kEN and disjoint Al, ... ,AkEB the distribution of
As shown in [5], a simple
point process
~
is symmetrically distributed with respect to some diffuse
(non-atomic) measure
00
iff
~
is a mixed Poisson or sample process.
case of random measures and diffuse
00
with
ooS<oo, a canonical representation
was given in [6] in terms of a random variable a
diffuse mass of
~,and
whose atom positions prescribe the atom sizes of
~
~
0, prescribing the total
a (canonical) point process
In the particular case when
In
~,
(3
on
R'
+
= (0,00),
cf. (1.2).
S is a real interval and
is seen to be symmetrically distributed with respect to
is Lebesgue measure,
00
00
iff the corres-
ponding cumulative random process has interchangeable increments, so in this
case the theory of [7] and [8] applies.
However, a direct extension of the
results there to more general spaces is certainly not easy (cf. [6]).
Further-
more, certain substantial improvements are obtainable in the present case,
similar to those attained in [5], Theorem 3.1., when specializing the central!
limit theorem to non-negative random variables.
Finally, the present
mo~e
general framework suggests some interesting generalizations which seem rather
artificial on the line.
For these three reasons, the theory of weak and
strong convergence deserves its own treatment for random measures being given
here in Section 2, along with a new proof of the canonical representation in [6].
1
~
2
In Section 3, we extend a result by Slivnyak [13], Papangelou [10] and
myself ([5], Theorem 5.3) by showing that the class of symmetrically distributed random measures may be characterized by a simple invariance property of
the corresponding Palm distributions [4,5].
Furthermore, it will be shown
that a natural strengthening of this invariance condition will essentially
delimit the class of sYmmetrically distributed random measures with symmetrically distributed canonical point processes, (containing in particular all
homogeneous (with respect to w) additive [5] random measures).
Extensions of
Slivnyak's theorem in an entirely different direction have been given by
Kerstan, Matthes and others (see [9,4]).
Paralleling the exposition in (7], we shall now introduce four types of
symmetrically distributed random measures on
S.
Type I random measures are
by definition of the form
k
~ =
where
R+
L n. O
j=l J t j
kEN, tl, ... ,tkES, and the nj
with canonical point process
with a unit atom at
but with k =
00.
that
TI
(1.1)
are interchangeable random variables in
=
L 0n
j
j
[7]. Here 0 EN(S)
s
Type II random measures are also given by (1.1),
SES.
{t.}
We then suppose that
J
has no limit point in
are interchangeable random variables in
random measure
~ [7].
is the measure
In both cases we put
v = LOt.'
j
R
+
Sand
with canonical
Type III random
J
measures are given by
00
~ = aw +
for arbitrary
in
wEM(S)
wS
with
=
{L.}
J
with
L S.
j
J
(1. 2)
Lj
1, independent random elements
w, and random variables a
S with common distribution
independent of
L 13·0
j =1 J
<
00
~
0, 13
L ,L
l
1
2
, ...
~ S2~"'~0
The associated canonical point process
3
R~
on
is
13
=~
J
0 . .
S
e'
Finally, a Type IV random measure is by definition
J
conditionally homogeneous with respect to
w, additive and infinitely
divisible with L-transform (L = Laplace) given, for measurable
2
f: S -+ R ,
+
by
=
ywf
II
+
(l-e
-xf(s)
)~(dx)w(ds)
a. s.,
S R'
+
(cf. Lemma 3.1 in [5]).
Here
R'
is a random measure on
(V,TI) ,
(v,~),
(w,a,S)
s
and
0
with
f
x~(dx)
l+x
<
(w,y,~),
is a random variable and
00
a.s.
The canonical quantities
which clearly determine the distributions
if any.
For convenience, we introduce some further notation.
and measure
for measures
m on
S, we write (fm) (dx)
g(x) - (l+x)
distribution [2] will be denoted by
¥ and
w
-+
= f(x)m(dx)
For any function
and
mf ::
_ k
k
m (dx) = x m(dx), keN.
m on R+ we define
reserved for the function
write
~
S, will always be defined as above, with the same affixes
of the corresponding
as
+
~
wEM(S), y
-1
.
f
Jf (x)m (dx), and
S
The letter
g
.
e,'
is
Equality and convergence in
£. and -+d respectively.
We further
for vague and weak convergence of measures [7], and to
distinguish between the corresponding notions of convergence in distribution,
we use th e symb 0 1 s
iate
by
B and
AEB
2.
~
and ~.
~
~.
For
a,S,y, and
B and
as above, we often abbrev-
(Note that the present use of
A differs from that in [7,8].)
Finally,
in any fixed metric generating the top logy of
Convergence and Related Topics.
~
a,
IAI denotes the diameter of
S.
In the following theorem we give extensions
and partial improvements of the criteria for convergence in distribution, given
for random processes with interchangeable increments in [7], Theorems 2.2, 2.3,
4.1, 3.2, 4.2 and 3.3.
2
This obvious attribute will be suppressed in the sequel.
J
4
Assuming
Theorem 1.
~
to be of Type
III~
~ n w1 ~ if,
we have
('OT'
[
J"n
of Type
I:
\>
n
S +
00
\>
'
III:
n
and faT'
~
\>
..
-
r
III:
c
n
n
if the
atoms and with
respeatively~
~
(2.4)~
wS
=
00,
C
g7T
Ir n
V
+
w,
n
aPe of Type I or III and
0, then
while if the
we have
~
1 wd
-+ g!l.
(2.3)
n
1 wd !I.
rng lln -+g
wd
c gB -+ g!l. ,
n n
wd !I.
c g!l. -+g
n n
v
+w,
w Ic X. w ,
n n
v
w Ic +w,
n n
P{~~O} >
(2.2)
B
of Type
n
+ 0
without fixed atoms and with
(2.J)~
n
vn Ir n
~n
~
B
+00
IV:
Conversely~
n
(2.1)
w
to be of Type IV with
{c }
II:
w
+ w ,
n
~
W
+
Iv n S
w
while assuming
.
n
~
n
(2.4)
(2.5)
(2.6)
wd
-+ some
~
wi thout fixed
is of Type III and (2.1) aT' (2.2) holds
aT'e of Type
I~
II~
p{~S = oo} > O~
III
then
vd
-+
OT'
IV and
~
is of Type IV and
~
n
(2.5) or (2.6) holds T'espeatively for some {r } and/oT'
some
{c }
n
n
Note that pleasant criteria for convergence towards mixed Poisson and
sample processes may be obtained by
~ecializing
to point processes.
In
particular, Theorem 3 of Benczur [1] follows by combination with Theorem 5.2
in [5].
Proof.
The sufficiency part will only be proved in the case I
remaining cases being similar, so suppose that (2.1) holds.
nn1, ... ,nnk
be the atom positions of 7Tn
n
us first assume the 7T
to be non-random.
n
+
For
III, the
nEN, let
taken in random order, and let
From (2.1) we get
5
in particular the
7T
2
n
nnj
are uniformly bounded, and we may thus conclude that
~. 13 2 (cf. Theorem 5.2 in [2]).
to random
e"
By Theorem 5.1 in [2], these results extend
in the sense that
(2.7)
Now define the random processes
X (t)
n
and let
==
n
In.
t nJ
j:O;k
by
in 0[0,1]
X
tdO,l],
n
X be a random process in 0[0,1] with X(O)
increments, possessing canonical random elements
=
° and with interchangeable
BR, 0, 13, in the sense of [7].
X ~ X in the Skorohod J topology [2].
l
n
and wClA. == 0,
~onsidering arbitrary mEN and Al, ... ,AmEB with Al c .•. c A
m
J
j == 1, ... ,ill, we get by (2.1) v A./k -7- wA.
j == 1, ... ,m, so
By Theorem 2.2 in [7], (2.7) yields
n J
n
J
(Xn (v nAl/kn ), ... ,Xn (v nAm/k))
n
m
in R+
by Theorem 5.5 in [2], since
d
-7-
(X(wAl), ... ,X(wAm.))
X has no fixed jumps [7].
But by inter-
changeability, this is equivalent to
(2.8)
Taking differences, it is seen that (2.8) remains true even without the
Festriction Al
c ..• c
Am' and since the class U == {AEB: wClA
a DC-ring in the sense of [5], satisfying
1.1 in [5] yields
and
~aA ==
~ n ~ ~. Finally by (2.7),
~ n ~ ~ follows as asserted.
°a.s.
= O}
is clearly
for any AEU, Theorem
e:·
6
Conversely, suppose that the
some
are of Type I and that
~n
with
~ { s}
Then
l
n
7T R =
.~ n 5 & ~5
<
00
= ()
P{ ~;i: O} > 0
a • s ., 5 c S;
(2.9)
l
l
a. s., so {(7T R, 7T )} is vaguely tight in R
n
n
+
and hence by Theorem 5.1 in [2] and the point process nature of the
l
n
is even weakly tight.
{7T }
Furthermore,
{v Iv 5}
n
x
M(R+),
TIn'
is automatically weakly
n
M(S), (the bar for one-point compactification).
relatively compact in
follows that any sequence
N'
c
N must contain some sub-sequence
N"
It
satis-
fying the conditions
wd some
TI 1 --?-
B .
ln M(R ),
nEN" ,
(2.10)
v Iv 5 ~ some w in M(S), nEN".
(2.11 )
n
n
To prove that
sup{v 5: nEN"'} <
n
n
5, we would get
d
(2.12)
for some sequence N'"
00
n
in
N'"
{s}
n
c
5 with
v n{sn}
2:
1, nEN "' .
But since
n
again contradicting (2.9).
This proves (2.12).
i
0
= ~5,
By the sufficiency of (2.1), we
from (2.10) - (2.12)
~n ~ some 1; in M(s), nEN",
where
is a random measure in
w,a,S •
But (2.13) implies both
by combination
we get
N" .
had no limit point
0 by (2.9), we may conclude from interchangeability that ~ 5
now obtain
c
~n ~ 0 ~ ~ contrary to (2.9),so we may assume the exist-
ence of some converging sequence
~n{sn} -+
nEN",
00
v -atoms corresponding to
If the set of all
in
n
v5-+
n
suppose on the contrary that
+
5 of Type III and with canonical quantities
~n 5
& ~5
£i 1;5 and
~
n
5 =
~
n
-d
5 -+ 1;8, and hence
1;5 = 1;5, so e.g. by considering the expectation of e -1;8 -e -1;5
d
-
1;(5\8) = 0 a.s., and finally
also true in
(2.13)
M(5), and that
w(S\5) = O.
2:
0,
It follows that (2.11) is
~(~ 1; on 8) is of Type III with the same canonical
7
quantities.
Furthermore,
must be diffuse by (2.9), and sow,a and S
W
,are a.S. uniquely determined by the diffuse component and the atom sizes and
positions of
The proof of (2.1) may now be completed by applying
l;.
'Theorem 2.3 in [2].
A similar argument proves the necessity of (2.2).
We next consider the necessity of (2.4) when the
.l;
n
~ some
l;
with the stated properties.
l;n
are of Type II and
Choosing compact sets
C.t S with
J
iC. c C? l' P{l;C.>O} ~ 0 and l;aC. = 0 a.s., jEN, we get l; ~ l; in each M(C.),
J
J+
J
J
n
.J
jEN, so by the necessity of (2.1), there exist diffuse measures w ,w ,···E'M(S)
l 2
such that
v Iv C.
n
Furthermore,
r
n
=
v C
n 1
+
00
n J
w
+
w.
J
in M(c.),
J
jEN.
and the restriction of
'
distributed with respect to
w.•
J
to
l;
In particular then
J
r Iv c.
n
is symmetrically
C.
n J
+
wJ'C l
r 0,
50
vir
n n
=
(v
n
Iv
C.)(v C./r ) ~ w./w,C
n J
n J
n
J
in M(C.)
J
J 1
jEN.
w./w.C , jEN, are all restrictions of some conunon diffuse
J J l
v
measure wEM(S) such that v Ir + w in M(s). Moreover, l; is symmetrically
n n
since otherwise
distributed with respect to w, so we must have wS =
Thus the measures
00
~s
<
00
a.s.
so for large
Now let
AEB be such that
wA > 1 and
waA
=
O.
Then v Air
n
n
+
wA
> 1,
n
(2.14)
by interchangeability.
This proves tightness in
R+
of the sequence of leftmost
members in (2.14), and also, by interchangeability, of the sequence
nEN
~f
random elements in
sub-sequence
00
R+,
Hence, given any sequence
(2.15)
N'cN, there exists some
N"cN' for which (2,15) converges in distribution, and by Theorem
_
..
jEN,
~
1.3 in [7] we get
~n'
*r
n -+- some
~,
n
neN".
In the particular case of non-random
r
it follows by Theorem 3.1 in [5] that
extends to general
[7]).
~n
n
g~
1 w
-+- some
n
g1\., nEN", and this
by randomization, (cf. the proof of Theorem 3.2 in
Hence (2.4) holds for nEN", and in particular i t follows by the suffic-
iency part that
is of Type IV with canonical quantities
~
as in the proof of Theorem 3.1 in [7], it is seen that
and so (2.4) holds for nEN by Theorem 2.3 in [2].
W,y,A.
and
y
Proceeding
A are unique,
The necessity of (2.6) may
be proved by similar arguments.
We finally consider the necessity of (2.3) when the
~
r
vd
n
n
-+
some
with the stated properties.
~
are of Type I and
Proceeding as above, we get
v
= vn C-+-oo
, and further v Ir -+- some w, where w is diffuse with wS =
1
n n
and such that
~
is symmetrically distributed with respect to
liminf c- l
n
n+oo
If
~
"'n
N' is an
arb~trary
converges as
n -+-
00
= liminf
n+oo
sub-sequence of
through some
N"cN',
v Sir
n
n
~ wS
=
w.
00
Moreover,
00
N, it follows as before that (2.15)
and by comparison of Theorems 3.2
and 4.1 in [7], it is seen that this remains true with the n . of (2.15) ren]
placed by some
n~j
which for fixed nEN" are interchangeable random variables
with canonical random measure
~
n
= ~n Iv n S = cn ~n Ir.
n
= cn g~ln ~
Hence
some g1\. in M(R ), nEN".
+
The remainder of the proof is similar to that of (2.4).
A similar argument
proves the necessity of (2.5).
We shall now show how the canonical representations of symmetrically distributed random measures given in [6] for bounded
1.
w, may be deduced from Theorem
(See also Theorem 5.1 in [5] and Theorems 2.1 and 3.1 in (7].)
9
.diffuse.
~
Let
Corollary 1.
~
Then
be a random measure on
P{~~O}
Assume that
we may suppose that
disjoint sets
be
wEM(S)
is syrrmetricaZZy distributed with respect to w iff
is of Type III or IV (depending on whether
Proof.
Sand Zet
or wS = (0).
00
We first consider bounded
>0.
wS = 1.
wS <
For each
nEN, divide
I .EB with
E;,
S, in which case
S into finitely many
wI . > 0 and waI
= 0, j=l, ... ,k , and such that
nJ
n
nJ
nJ
{I n+l,j } is a refinement of {I .} for each n and maxlI .1 -+- O. Put
nJ
nJ
J
r = nl (min wI
and choose v = lOt EN(S) with v I . = [r wI .J. An easy
n
n
nJ
n nJ
n nJ
k nk
0
0
)
calculation yields
v I .
1
1 _ - < n nJ
n
r wI
n
j=l, ... ,k,
n
nJ
which clearly remains true with
sets among
-< 1 ,
0
nEN,
replaced by any non-empty union U of
nJ
For such U,
Inl, ... ,I nk .
I
0
n
v U
-E.-_. wU
v U - v SwU
n_ _.-:n
~ 21_
__1
r wU
n
v S
n
and in particular,
~n
with
n
n
= (v S)
nJ
n
are of Type I, and ~
A .EB
nJ
v Iv S ~ w.
wA
-1
0
n
For
~
v U
2
n
r
n
of
(wUOl, ... ,wU Ok ) implies
~
and the fact that
~
n
2
v S
n
1
r wS n
.
<
4
n
(20 16)
S into disjoint sets
\'
, J=l, .•. ,v S, and put ~ = L ~A oOt
• Then the
n
n
j
nJ nj
~ ~ follows from (2.16) by Theorem 1.1 in [5J and
Unl, ... ,Unk , nEZ+, (wUnl, ... ,wUnk )
(Wnl"",~Unk)i (~U01"",~UOk)'
Un +~
implies
from the converse part of Theorem 1 that
unbounded
+
nEN, we next divide
the easily verified fact that, for disjoint
-+-
wU -
1
~U
n
~
-+- 0
a. s.)
(Use the symmetry
We may now conclude
is of Type III.
In the case of
S, we may again use the converse part of Theorem 1, now with the
chosen as restrictions of
~
to some suitable sequence of bounded sets.
e-
10
For the applicability when
A ,A 2 , ... EB
l
with
P{~s
wA k
= co}
>
wS
= 00,
note that by Fatou's lemma, for disjoint
1 and for sufficiently small
~ P{~Ak > E
i.o.}
~
limsup
0,
E >
P{~Ak > E} >
0.
k-+oo
In [7], Theorem 5.3, it was shown that the distribution of any random
process with interchangeable increments is uniquely determined by that of its
restriction to any fixed sub-interval.
Obviously, this result generalizes to
the case of random measures, yielding alternative convergence criteria
in Theorem 1.
In the particular case of non-random canonical quantities
(allowing interpretations in terms of sampling from finite populations), we
may obtain still simpler determining (and therefore also convergence deter-
··e
mining) classes ([2], p. 15).
In fact, sYmmetrization in Lemma 11.2 by Rosen
[12] (cf. Theorem 12.1 in [12] and Theorems 4-5 in [3]) yields the
Proposition 1.
that
Let
~
B is non-random.
be a random measure on
If w is
~A
uniqueZy determined by that of
known~
S
III~
of Type
and suppose
then the distribution of
for any fixed AEB with
°
<
wA
~
&s
<
wS.
Note that the corresponding statement for Type IV random measures is also
true.
B (or A) to be
For simple point processes, we need not even assume
non-random
([5], Theorem 5.2).
We conclude this section by considering extensions and partial improvements of the variational and ergodic results given for random processes with
interchangeable increments in [8], Theorems 5.1, 6.2, 6.3 and 6.4.
Clearly,
nn should now denote a partition of S or of some Sn EB into disjoint
measurable sets
For any mEMeS), we write
IT
n
m
II
III n I
In analogy with [8], we further define
=:
max
00
2
(II w) R, and we say that
n
~andom
J
Bn
"
= \ (w/\.)
<.
2
11 J
J
,
n
wS
=:
measure corresponding to a partition of
p > 0, while
nJ
{II } is nested if it proceeds by successive Tcfinc-
For Type IV random measures with
ments.
1][n /22
lIl/\ .,
•
denotes the canonical
~p
00,
S into sets of w-measure
corresponds to the restriction of
~
to
S EB.
n
Using these
notations, we may state the strong counterpart of Theorem 1 as follows.
Theorem 2. If
~
is a Type III random meaSU:l'e and if IT l , IT 2 , . .. are parti tions
of S which are -either nested with IITnioo
+
I/ITnl~
or satisfy
0
<
00
,
then
n
(IT ~)l ~ B a.s. in M(R )
n
On the other hand, if
~
of Type IV with wS
~s
(2.17)
+
=
00
then
a.s. in M(R).
wS
+
n
00,
and for any
f: R + R ,
+
+
fB /wS ~ fA in M(R ), a.s. within {Af
n
n
+
For such
{S}, Zet the
n
(2.18)
<
oo}.
(2.19)
II be partitions of S , nEN, satisfying
n
n
2
lITn 12/wSn
+
IIIT /2 /(wS )2
0,
n
n 2
n
<
00
•
Then
1
g(IIn ~) /wSn
It
w
+
gA
should be noticed that the monotonity of
{S }
n
truth of the statements involving (2.19) and (2.20).
ness of
(2.20)
a.s. in M(R+") .
is essential for the
Similarly, the nested-
{IT } is essential for the truth of (2.17) in the case
n
However, the nestedness can be dispensed with if we assume
and change the definition of
IITnioo
to
/IT n I
00
= max IAnJ. I .
•
J
w to be diffuse
e-
-e
12
Proof.
To prove the first assertion, we may clearly assume that
B is
1I01l·-
random and that
wS = 1.
The probability that two particular atoms lie in the
same set of
is then
2
Inn 12
nn
event is non-increasing in
vided Inn 100
O.
+
/nn 100
.
If the
n
n
arc nested, then this
n, so it can a.s. only occur finitely often pro-
r Innl;
If
::;
<
00 , the same statement follows by the Borel-.
n
Cante1li lemma.
The extension to any finite set of atoms being immediate,
it follows easily that
X(3
nn ~
a.s. in N(R') .
Since
+
identically, this completes the proof of (2.17).
The assertion involving
(2.18) is essentially equivalent to the converse part of Theorem 3.1 in [5].
The remainder of the proof is easy, given the exposition in [8].
3.
Invariance of Palm Distributions.
In this section we shall show that
the defining property of symmetrically distributed random measures is closely
related to some other symmetry properties, expressible in terms of Palm
distributions.
Just as for point processes in [5], the latter are defined for
~
arbitrary random measures
measures
~s'
SES,
E~EM(S)
with
as the distributions of random
satisfying, for any f: M(S)
=
(Cf. [4] for existence.
Ef(~) ~
(ds)
E~(ds)
+
R+,
(3.1)
a.e. E~.
, SES
For the related concept of Campbell measure, see [9].)
We refer to [5] for the notion of mixed sample processes and for the abbreviMS(w,~).
ation
A measure is said to be degenerate if all its mass is concen-
trated to one single point.
Let
Theorem 3.
~
s
- ~ {s}6 )
s
s
d
~
= some
be a random measure on
(n,~)
S with
independentZy of s a.e.
(exaept possib Zy for a. s. degenerate
~)
and
~
E~3
E~EM(S).
iff
E~
Then
(~
s
{s},
is diffuse
is symmetriaa Uy distributed
ewith respect to
E~.
with respect to
E~,
(i)
~
In this case, (n,s) is also symmetrically distributed 3
and furthermore,
is of Type III with
a
n
is independent of
s iff either
a.s. and S a mixed sample process, or
= 0
A = pM a.s. for some random variable
(ii) ~ is of Type IV with
p and some
non-random MEM(R).
+
For the particular case of point processes, a slightly stronger assertion
was proved in [5], Theorem 5.3, (see also [10]).
A (very) special case of the
situation in (ii) was discussed by Port and Stone in [11], Example 2.
As will be seen from the proof, the distributions of
and (n,s) are
~
related, in case of symmetry, by either or both of the relations
Ef(n,B)
=
Ef(n,A)
=
1 E
EBR
B-xo )B(dx) ,
x
fR f(x,
(3.2)
1
EiiR E J f(x,A)A(dx) ,
(3.3)
R
f: R+ x M(R +) -+R +'
for arbitrary
canonical random measures of Z;.
distribution
A = pM
EB/EBR, while
a
M on
some random variable
= 0
R+, then
p
~
a
If
for some random variable
ability measure
where
0 with
p
B
= 0
ao
=
and
Al
+ S and
O
S
£!.
MS(ES,<jJ), then n
and 13Q. MS(ES,-<jJ').
~
0 with
<jJ
and some prob-
M while
A = pM
for
-<jJ'.
We finally remark that the cases (i) and (ii) where
ent are not so far apart as they may appear.
has
On the other hand, if
L-transform
n has distribution
L-transform
A denote the
nand
Z;
are independ-
In fact, as may be seen from
Theorem 1 (or rather from its proof), the random measures satisfying (i) or (ii)
3:
This means of oourse that the distribution of (n,Z;Al, ... ,sAk ) only depends
on
(E~Al,
...
,E~Ak).
14
constitute the closure with respect to convergence in distribution in the
vague topology of the class of random measures satisfying (i).
For the proof of Theorem 3, we need a lemma of some independent interest.
Let
Lemma 1.
be a random measure on
~
be a random element in
~/~S.
Then
given that
S which for given
~
~
1
0" iff
~
~S <
a.s. and let
00
T
1 0 has conditional distribution
Tis conditionaUy independent of
(n" Z)
=
(~h}, ~ - ~h}o ),
T
is symmetrically distributed with respect to some
degenerate~)
diffuse (except possibly for a.s.
~
S with
In this case"
wEM(S).
~
(n,~)
is conditionally symmetrically distributed with respect to
~
canonical random measure
~
E[f(n,B)
I~
~
of
B
satisfies" for
f: R
+
M(R )
x
+
w" and the
+
R ,
+
1 0] = E[B~ J f(x, B-xox)B(dx)/B 1 0].
(3.4)
R
Note that, in the particular case of point processes,
positions chosen at random.
T
=
3.
S, SES
a. e. PT
-1
,
T
is one of the atom
Clearly the conditional distributions of
~,
given
here play the role of the Palm distributions in Theorem
Though perhaps at least as natural from the point of view of applications,
they do not behave quite as well mathematically.
A related type of competitors
to the Palm distributions was considered by Slivnyak [13] for stationary point
processes.
~
Proof of Lemma 1.
Suppose that
T
with distribution
w, given that
~
and
~ S2~
Sl
...
of
and
1
the atom sizes of
o.
~
~
~
Since
is conditionally independent of
Let
with distribution
B as fixed.
be the total diffuse mass
, and put S = L
j
~
(n, B) depends measurably on (n, ~) ,
(n,Z)
CI.
T
°f3 J• '
B
= Cl.0 0
T ,T
l
2
+
Sl.
is even conditionally independent
w, given any (n,B) 1 0, so we may consider
Now define
~
(n,~)
n
, ... as the atom positions corresponding
IS
-to 131'13 2 "", taken in random order within sets of equal 13 j .
13 1
=... = 13 n
13
>
and choosing
+
n l
with either of
11
we may clearly identify
Tl, ... ,T , so i t follows by induction that
n
~ -
are independent of
I
13.0
j=l J
,.., ,..,
~
~,
of
If
a > 0,
T, given
~,
> 0, the last state-
11
we next put
11 =
Then
O.
coincides with the normalized
and this remains conditionally true, given
,..,
(11,l;) since
~
depends measurably on l;, so we get
~ =
w a.s.
then
a
13 2
=0
=0
a.s. since
a.s., and hence
Conversely, if
diffuse
aw
~
If
w{s} > 0
is the diffuse component of
~
~.
B,
For given
is therefore conditionally symmetrically distributed with respect to
and this clearly remains true unconditionally.
so
n
and mutually independent with distribution
the conditional distribution of
,..,
Tl, ... ,T
Tj
ment remains true for arbitrary n.
diffuse component
T
n
Since the same argument applies to any choice of
w.
~
= 13 1 ,
Assuming
w,
for some
SES,
Furthermore,
= 13loTl.
is symmetrically distributed with respect to some
wEM(s), it follows by Corollary 1 that, for given (n,B) F 0, T is
,.., ,..,
conditionally independent of (11,l;) with distribution
remain true unconditionally.
distribution
Since, for given
B F 0,
w, and this will clearly
11
has conditional
B/BR, (3.4) follows easily by the same corollary.
Proof of Theorem 3.
Since forming the Palm distributions and restricting to
a sUb-space are interchangeable operations, we may assume that
when proving the first assertion.
any
f: M(S)
x
S
+
Let
R+ , uER + and SESe
~S <
00
a.s.
T be defined as in Lemma 1 and consider
By Proposition 2 in [4], which clearly
carries over to arbitrary random measures,
e-
.-
16
E[f(~,T); ~SEdu, TEds] = E[f(~,s)P{TEdsl~}; ~SEdu]
= E[f(~,s) ~(ds);
~SEdu] = !u E[f(~ s ,s); ~ s SEdu]E~(ds),
u
so by the chain rule for Radon-Nikodym derivatives, for (u,s) E R
E[Hds);
S a.e.
x
+
~SEdu] ,
E[f(~,T)I~s = u, ~
=
5
+
T
Now if the distribution of
(~
s
{s},
~ -~
5
s
conditionally independent of T, given
will not change the definition of
~S
= u.
(~{T},~-~{T}O
T
Since conditioning on
T in terms of
~,
~S
=u
Lemma 1 applies, and so
~) W
u
EM(S).
uER, by the assumed invariance,
+
J
P{~ SEdu}E~(ds)
5
= p{~ 5 SEdu}E~A
,
so by the chain rule
and we get
~
1
__
__ E ~A; ~SEdu]
E~A
-;:::+.~--=::~-r-----"~~..;.--..~ = Et"S
U E [?:~ A I~?: S u]
E ~S; ~SEdu
~
= -
W
u
= E~/E~S,
(3.5)
) is
A
UA
+
is conditionally sYmmetrically distributed with respect
E[~A;~SEdu] =
W
R ,
+
s, then
5
to some normalized diffuse (except possibly for a.s. degenerate
and
M(S)
x
{s}o ) is a.e. independent of
so is the right hand side of (3.5), and it follows that
for any AEB
= u],
= U,T = 5] = E[f(~ 5 {s},~ s -~ S {s}o S )I~ S S = u].
E[f(~{T}, ~ - ~{T}O )I~S
~
S
In particular, this yields a.e., for f: R
(cf. Lemma 5.1 in [5]).
we may conclude that
E[f(~ ,s)l~ S
independently of
u
~
0 a.e.
is even unconditionally sYmmetrically distributed.
P(~S)
-1
, proving that
But
17
Conversely, suppose that
quantities
wand
B, where
~
is symmetrically distributed with canonical
w is diffuse and
0 < EBR <
For
00.
f: R+
R+ ,
-+
we get
E[~(ds)f(~8}]
= E[E(~(ds)IBR)f(BR)] = E[BRf(BR)]w(ds),
so by (3.1),
" 8)
Ef( ~s
= ~[BRf(BR)J
EBR'
and in particular, the distribution of
~
s
8
Se 8 a.e. E"~,
(3.6)
is a.e. independent of
(~
Furthermore, by (3.5) and Lemma 1, the conditional distribution of
~ -~
s
s
~
{s}o ), given
s
s
8, is a.e. independent of
is true for the unconditional distribution.
s.
s
{s},
s, so by (3.6), the same thing
From (3.5) it is also seen that
(n,~) ~ (~ s {s},~s -~ s {s}o s ) is symmetrically distributed with respect to w,
and by combination of (3.4) and (3.5),we get for
A-
I
E [f ( n ,B) ~ 8
s
= u] = B~
E[
I
f: R+
M(R+)
x
f(x,B-xo )B(dx)IBR
x
R
u >0
-+
e"
R+
= u],
a.e. P(~ 8)-1 ,
s
which yields (3.2) when inserted in (3.6).
To prove (3.3) when
and
A, w8
=
let
00,
~
Band
is symmetrically distributed with canonical
B correspond to the restrictions of
respectively to some Ae B with wA
a..
= ty
while
f3
=t
> O.
I
is a Poisson process with intensity
f(x,a.,f3-o x )B(dx)
R
=E
I
and
~
t>.. [7], so by (3.1)
x
R+
x
N(R +')
f(x,a.,f3- ox )Xf3(dx) +E[a.f(O,a.,f3)]
R
= I.
Ef(x,a.,f3 -0 )xt>..(dx) + E[tyf(O,a.,f3)]
x x
R
=tJ
R
~
If A is non-random, we have
and the remark following Theorem 5.3 in [5], we get for f: R+
E
w
Ef(x,a.,f3)x>"(dx) + tE[yf(O,a.,f3)]
= tE
ff(x,a.,S)A(dX),
R
-+
R+
IH
and this result extends by conditioning to arbitrary
we get by (3.2) for
f: R+
M(R+)
x
~
AtS
we get
Since
EBR
=
(3.7)
f(x, B/t)A (dx) .
by Theorem 2
tEAR,
R+
Ef(n,B/t)
Letting
A.
BIt ~ A and
BIt
~ A a.s. in M(R + ), so
f, (3.3) follows from (3.7) by repeated dominated
for bounded and continuous
convergence, and we may extend (3.3) successively, first by monotone convergence to indicators of open sets (cf. Theorem 1.2 in [2]), then by Dynkin's
theorem to arbitrary indicators, and finally by linearity and monotone convergence to arbitrary
Now suppose that
o<
E~S
ing
~
<
00,
f.
s
nand
are independent.
We may assume that
since otherwise the proof may be reduced to this case by restrictBy (3.1) and (3.2), we get for any t
to bounded sub-spaces.
e
1
= EBR
-xt
~
0 and
f(B-xc )B(dx)
x
JR e-xtEf(Bx -xc x)EB(dx)
,
so if 0 < Ef(B) < 00, we obtain
Ee
-nt
=
Je- xt Ef(Bx-xc x )
R
Ef(B)
EB(dx)
EBR
Comparing this with the formula obtained for f - 1, it follows by the uniqueness theorem for L-transforms that
Ef(B -xc)
x
x
= Ef(B),
x
~
0 a.e. EB.
19
Turning to an
a.e.
f: R
+
x
N(R ')
+
R
-+
+
and using (3.1), we thus obtain for x > 0
ES
E[B (dx) f (a, S-ox)]
Ef (a, S)
E[S(dx)f(a,s-ox)]
= --....,E=B,-;(~dx-:)-~- = ---;::E"::"S"7"(d'x')---
(in an obvious notation), and also, provided
From now on, we may assume that
restrictions of
ESR
< 00,
Ef (ax' Sx- ox),
(3.8)
Ea > 0,
since otherwise we may consider the
to compact sub-intervals of
(3
=
R'+ .
Proceeding as in the
proof of Theorem 5.3 in [5], we may conclude from (3.8) that, for given
v = SR,
vES/Ev.
13
is conditionally a sample process independent of
At this stage, we may assume that
Ea
and
otherwise either (i) or (ii) is trivially satisfied.
then get for any t
~
0 and
Ev
a
with intensity
are both> 0, since
By (3.8) and (3.9), we
sE[O,l]
E{e -at s v-I E[S(dx) Ia,v]}
ES(dx)
=
E{e-atsv-lVE[S(dx)]/Ev}_ E[e-atvs v - l ]
ES(dx)
Ev
so
and hence by the uniqueness theorem,
= E[vsv-ll a ]
Ev
a.s.,O=:;s=:;l.
(3.10)
Assuming these expectations to be calculated from some family of regular
conditional distributions of
v, given a, (3.10) extends by continuity from
any countable dense sub-set of s-values, so we may take the exceptional P-null
set in (3.10) to be independent of s.
Writing
and
20
c
= Ev/Ea,
and since
we get a.s. the differential equations
~
a
=1
(1)
a.s., we obtain a.s. the unique solutions
~a(s)
= e-ca(l-S),
0 ~ s ~ 1 ,
v, given a, are a.s. Poisson-
showing that the conditional distributions of
ian with means ca.
But this is merely another way of expressing (ii).
Conversely, assuming
to be such as in (i), we get by (3.1), (3.2)
~
and Theorem 5. 3 in [5], for any t
Ee-
nt Bf
-
= EBiR
E
I
~
0
and f: R'+
-+
R+
exp[-xt-(B-o )f]xB(dx)
x
R
=
so
nand
1_
EB1R
I e- xt
R
E exp[-(B -0 )f]xEB(dx)
x x
B are indeed independent with the asserted distributions.
Similarly, for
~
as described in (ii), we get by (3.4) for any t
~
0 and
f: R -+ R
+
+
Ee- nt - Af
=
1_ E
EpMR
I e-xt-pMfpM(dx)
= E[pe- PMf ]
R
I· e-xtM(dx)
,
R
in conformity with our assertions.
This completes the proof of Theorem 3.
It should be observed that, in the proof of the second assertion, the
crucial point is to show a
must be a.s. zero if
~
is not of Type IV.
21
In fact, assuming
Ee - nt -
Af
=
s
to be of Type IV, (3.1) and (3.3) yield for any
1
E
EAR
f e-xt-AfA(dx)
R
so if
n and
E exp[-A f]
x
~
=J
e-xtE exp[-A f] EA(dx)
x
EAR
R
are independent, it follows by the uniqueness theorem that
= E e- Af ,
x ~ 0 a.e. EA, and hence that A ~ A a.e.
x
Arguing as
in the proof of the first assertion in Theorem 3, it is not hard to see that
this implies
MEM(R ).
+
A = pM
for some random variable
p ~
0 and some non-random
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DEPARTMENT OF MATHEMATICS, CHALMERS UNIVERSITY OF TECHNOLOGY
..
..
AND THE UNIVERSITY OF GOTEBORG, FACK, S-402 20 GOTEBORG 5,
SWEDEN
(Current address:
DEPARTMENT OF STATISTICS, UNIVERSITY OF
NORTH CAROLINA, CHAPEL HILL, N.C. 27514 (U.S.A.))