Symmetries on Random Arrays and. Set-Indexed Processes. Olav Kallenberg 1. Received July 22, 1991; revised March 16, 1992. A process X on the set N- of ...
Journal of Theoretical Probability, Vol. 5, No. 4, 1992
Symmetries on Random Arrays and Set-Indexed Processes Olav Kallenberg 1 Received July 22, 1991; revised March 16, 1992 A process X on the set N- of all finite subsets J of N is said to be spreadable, if (Xpj) d (X~) for all subsequences p = (PI, P2,...) of N, where p J= {pfi j ~ J}. Spreadable processes are characterized in this paper by a representation formula, similar to those obtained by Aldous and Hoover for exchangeable arrays of r.v.'s. Our representation is equivalent to the statement that a process on is spreadable, iff it can be extended to an exchangeable process indexed by all finite sequences of distinct elements from N. The latter result may be regarded as a multivariate extension of a theorem by Ryll-Nardzewski, stating that, for infinite sequences of r.v.'s, the notions of exchangeability and spreadability are equivalent. KEY WORDS: Exchangeable and spreadable processes; measure-preserving transformations; conditional independence and coupling.
1. B A C K G R O U N D ,
MOTIVATION,
AND MAIN RESULTS
A n infinite s e q u e n c e X = (X1, X2,... ) of r a n d o m v a r i a b l e s (r.v.'s) is said to be exchangeable, if
Xop~(Xpl,~p2,... ) d ( X l , X 2 , . . . ) : X
(1.1)
for all (finite) p e r m u t a t i o n s p of N = {1, 2,...}, a n d spreadable if Eq. (1.1) h o l d s i n s t e a d for all s u b s e q u e n c e s p of N, so t h a t the s u b s e q u e n c e s of X h a v e all the s a m e d i s t r i b u t i o n . R y l l - N a r d z e w s k i (a6) i m p r o v e d de F i n e t t i ' s c e l e b r a t e d t h e o r e m b y s h o w i n g t h a t those t w o n o t i o n s are in fact e q u i v alent, so t h a t e v e n s p r e a d a b l e s e q u e n c e s are m i x e d i.i.d. (cf. Kingman~15)).
1 Departments of Mathematics, Auburn University, 120 Mathematics Annex, Auburn, Alabama 36849-5307. 727 0894-9840/92/1000-0727506.50/0 9 1992 Plenum Publishing Corporation
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Kallenberg
The result can be phrased in terms of stopping times or martingales, and in continuous time it leads to some important connections between exchangeability theory and random time change (cf. Kallenberg 'n} n
i=1
Lemma 2.8. Consider for some d E N a separately exchangeable array X = ( X ~ ) , where I c { 1 ..... d} while m E N d. Then the array X ' = X l'-'d satisfies X'
L[
(2.7)
(x\x')
~(z')
P r o o f For d = 1 (and 2) the statement is proved in HooverJ 9) Proceeding by induction, we assume the statement to be true for d - l, and consider a d-dimensional array X. Define a (d-1)-dimensional array Y= (Y~) with values in R ~, by putting
y1m = { X m*, k,...,,~ yx~ {d}., k E N } ,
Ic{1,...,d-1},
mEN a-I
Writing Y ' = yl,...,d-~ and Z " = X ~''d 1, we get
(x',x")=r'
]_I (r\Y')=x\(x',x") .~( r')
and since 5~
c 5O(X') v a ( X " ) c a ( Y ' ) , it follows by Lemma 2.1(c) that
x'
(2.8)
LI ~(y'),x"
Next we apply the result for d = 1 to the sequence in R ~, Zk=(Zk,
Xm, " m E N d-1 },
kEN
to obtain Z'=Z'
LI
z"=x"
~(z')
Since clearly Y ( Z ' ) c 5O(X') c a(X'), it follows by Lemma 2.1(c) that x'L[
x"
..~(x')
Combining this with Eq. (2.8) and using the same lemma, we get Eq. (2.7). In subsequent sections, we shall only need the following corollary.
Symmetries on Random Arrays and Set-Indexed Processes
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L e m m a 2.9. Consider a separately exchangeable array X = (X~m) as before with independent entries, fix a function f : R2~--+ R, and define the arrays
X',,,={X~;tlI
