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Predictable sampling for partially exchangeable arrays
ARTICLE IN PRESS
Statistics & Probability Letters 70 (2004) 95–108 www.elsevier.com/locate/stapro
Predictable sampling for partially exchangeable arrays B. Gail Ivanoffa,1, N.C. Weberb, a
Department of Mathematics and Statistics, University of Ottawa, P.O. Box 450 Station A, Ottawa, Ont., Canada K1N 6N5 b School of Mathematics and Statistics, F07, University of Sydney, Sydney, NSW 2006, Australia Received 16 March 2004; received in revised form 15 June 2004
Abstract New predictable sampling theorems are developed for two natural extensions of the concept of F-exchangeability applied to separately exchangeable arrays and to jointly exchangeable arrays. Further we show that weak and strong F-exchangeability are in fact the same for certain ergodic arrays. r 2004 Elsevier B.V. All rights reserved. MSC: primary 60G09 Keywords: Separately exchangeable array; Jointly exchangeable array; Adapted time; Spreadable array; Predictable sampling
1. Introduction Exchangeable sequences and arrays of random elements have been studied extensively since the seminal work of De Finetti (1929, 1930). A finite or infinite sequence is said to be exchangeable if its distribution is invariant under permutations that leave all but a finite number of indices fixed. There is an extensive literature on exchangeable and related arrays. Aldous (1985) and Kallenberg (2002) provide excellent general coverage of the theory of exchangeability. Corresponding author. 1
E-mail addresses: [email protected] (B.G. Ivanoff), [email protected] (N.C. Weber). Research supported by a grant from the Natural Sciences and Engineering Research Council of Canada.
0167-7152/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2004.08.005
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Kallenberg (1982) introduced the concept of conditional exchangeability for sequences by relating exchangeability to a filtration F to which the sequence is adapted, and in Kallenberg (1988) proved a predictable sampling theorem: i.e. that the joint distribution of the first k elements of the sequence is the same as that of the sequence sampled at k distinct, F-predictable random times. Exchangeability is in general a stronger concept than spreadability: a sequence is spreadable if all finite subsequences of the same size have the same distribution. Kallenberg (2000) proves a weaker version of the predictable sampling theorem for conditionally spreadable sequences which requires the F-predictable random times to be strictly increasing. Nonetheless, there are situations in which spreadability is equivalent to exchangeability: in particular, if the sequence is infinite, see Ryll-Nardzewski (1957), or finite and ergodic, see Kallenberg (2002). Extending the ideas of exchangeability and spreadability to two-dimensional arrays leads to two different approaches. For the stronger separate exchangeability the joint distribution of the array is invariant under permutations of both rows and columns. If the joint distribution of the array is invariant under the same permutation of rows and columns then we say the array is jointly exchangeable. Separate and joint spreadability are defined analogously (see, Kallenberg, 1992; Ivanoff and Weber, 1996, 2004). The extension of conditional (F-) exchangeability (respectively, spreadability) to arrays is not entirely straightforward. In Ivanoff and Weber (1996) F-exchangeability of arrays was defined in two different ways and extensions of F-spreadability to arrays was explored in Ivanoff and Weber (2004) where the relationship between the different types of F-spreadability and twodimensional martingale structures was investigated. Predictable sampling theorems were proven for the various F-spreadable structures in that paper. The primary purpose of this note is to develop predictable sampling theorems for each of the four types of conditionally exchangeable arrays (see Section 3). In light of the corresponding theorems in Kallenberg (2000) and Ivanoff and Weber (2004), the appropriate statement of the relevant theorems can be anticipated; in some cases, the proof is a reasonably straightforward adaptation of one-dimensional techniques. However, in at least one case, a new and distinctly two-dimensional approach is required. We also explore conditions under which a spreadable array is exchangeable in Section 4. We will show that certain results for ergodic arrays in Ivanoff and Weber (1996) can be strengthened and that in these cases spreadability and exchangeability are equivalent.
2. Notation and definitions A finite or infinite random sequence x ¼ ðx1 ; x2 ; . . .Þ in a measurable space ðG; GÞ is said to be exchangeable if ðxk1 ; . . . ; xkn Þ¼D ðx1 ; . . . ; xn Þ
(1)
for any collection k1 ; . . . ; kn of distinct elements in the index set of x: Equivalently, X is exchangeable if for any permutation p of ð1; 2; . . .Þ leaving all but a finite number of terms
ARTICLE IN PRESS B.G. Ivanoff, N.C. Weber / Statistics & Probability Letters 70 (2004) 95–108
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unchanged, X p ¼ ðX pð1Þ ; X pð2Þ ; . . .Þ¼D ðX 1 ; X 2 ; . . .Þ ¼ X :
(2)
We say that x is spreadable if (1) holds whenever k1 o okn ; i.e. all subsequences of the same size have the same distribution. In the infinite case Ryll-Nardzewski (1957) showed that spreadability and exchangeability are in fact equivalent but as Kingman (1978) noted this is not the case for finite sequences. For clarity and notational convenience in what follows, given a c r array we shall generally assume that either both c; ro1 or c ¼ r ¼ 1: However, all definitions and theorems can be adapted in a straightforward way to include the case in which only one of c and r is finite. Let X ¼ ðX ij : 1pipc; 1pjprÞ be a c r array of random elements taking their values in some measurable space ðG; GÞ: X is separately exchangeable (SE) if for all moc; qor; m; qo1 ðX ik j h : 1pkpm; 1phpqÞ¼D ðX ij : 1pipm; 1pjpqÞ
(3)
holds for distinct, but not necessarily increasing, values i1 ; . . . ; im and j 1 ; . . . ; j q : (The array is separately spreadable (SS) if (3) is satisfied for all strictly increasing values 1pi1 o oim pc and 1pj 1 o oj q pr:) Alternatively, X is SE if and only if for any permutations p of ð1; . . . ; cÞ and s of ð1; . . . ; rÞ each leaving all but finitely many terms unchanged, X p ¼D X
and X s ¼D X :
(4)
Here, X p ¼ ðX pðiÞj : 1pipc; 1pjprÞ and X s ¼ ðX isðjÞ : 1pipc; 1pjprÞ: We have X ps :¼ðX p Þs ¼ ðX s Þp : The square array X ¼ ðX ij : 1pi; jpnÞ (n may be 1) is jointly exchangeable (JE) if for all finite mon ðX ik ih : 1pk; hpmÞ¼D ðX ij : 1pi; jpmÞ
(5)
holds for distinct, but not necessarily increasing, values i1 ; . . . ; im : (The array is jointly spreadable (JS) is (5) is satisfied for all strictly increasing values 1pi1 o oim pn:) X is JE if and only if for any permutation p of ð1; . . . ; nÞ leaving all but finitely many terms unchanged, X pp ¼D X :
(6)
In order to define conditional exchangeability for arrays, we assume that we are given an arbitrary filtration F ¼ ðFij : 0pipc; 0pjprÞ to which X is adapted (i.e. Fij Fhk if iph and jpk; and X ij is Fij -measurable, 8i; j). Roughly speaking, we will say that an array has a particular property in the weak F sense if, conditioned on Fjk ; the translated array yjk X :¼ ðX h‘ : j þ 1phpc; k þ 1p‘prÞ has the required property for all ðj; kÞ 2 f1; . . . ; cg f1; . . . ; rg: On the other hand, assuming that all necessary conditional distributions exist, suppose that the actions defining the property in question are carried out in such a way that the first j columns and k rows are left unchanged and consider the conditional distribution of the entire array given Fjk : Then we say that the array has the property in the strong F sense if the conditional distributions of the new array and the original are the same for all ðj; kÞ 2 f1; . . . ; cg f1; . . . ; rg:
ARTICLE IN PRESS B.G. Ivanoff, N.C. Weber / Statistics & Probability Letters 70 (2004) 95–108
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More precisely, the array c r X is weak F-SE if conditioned on Fjk ; yjk X is SE;
8joc and 8kor:
(7)
Similarly, the n n array X is weak F-JE if conditioned on Fkk ; ykk X is JE;
8kon:
(8)
The array X is strong F-SE if for all ipc and all jpr and any finite permutations p of ð1; . . . ; cÞ leaving ð1; . . . ; iÞ unchanged and s of ð1; . . . ; rÞ leaving ð1; . . . ; jÞ unchanged, X p ¼DjFij X
and X s ¼DjFij X ;
where ‘¼DjFij ’ denotes equality of the conditional distributions given Fij : The n n array X is strong F-JE if, for every ipc and any finite permutation p of ð1; . . . ; nÞ leaving ð1; . . . ; iÞ unchanged, X pp ¼DjFii X : The concept of strong F-exchangeability was introduced only for finite arrays in Ivanoff and Weber (1996); here we consider it for both finite and infinite arrays, and will see that in some sense it is the more appropriate analogue of the one-dimensional conditioning. It is clear that the strong F-properties imply the weak F-properties, and they are equivalent when F is the minimal filtration generated by the array. For general filtrations, the converse is not true: an example is given in Ivanoff and Weber (1996, p. 350) that is weak F-SE but not strong F-SE. Given the filtration F ¼ ðFij : 0pipc; 0pjprÞ denote the (one-dimensional) filtrations F1 ¼ ðF1i : 1pipcÞ and F2 ¼ ðF2j : 1pjprÞ; where F1i ¼ _rj¼1 Fij and F2j ¼ _ci¼1 Fij : Arguing as in Lemma 3.1.1 of Ivanoff and Weber (2004) we have the following result. Lemma 2.1. Let X be a finite or infinite array. X is strong F-SE if and only if the sequence of column vectors ðX i ; 1pipcÞ is F1 -exchangeable and the sequence of row vectors ðX j ; 1pjprÞ is F2 -exchangeable. Lemma 2.1 improves on Lemmas 2 and 3 of Ivanoff and Weber (1996) in a number of ways. First, conditional independence of F1i and F2j given Fij was assumed, but this condition is not needed. Second, the conclusion of Lemma 2 was that F1 - and F2 -exchangeability of the column and row vectors of an infinite array implies that X is weak F-SE; we see here that in fact X is strong F-SE. This stronger result was proven in Lemma 3 of Ivanoff and Weber (1996) in the case of finite arrays, but it was incorrectly stated subsequently that this is not a necessary condition. In fact, F1 - and F2 -exchangeability of the column and row vectors is not a necessary condition only in the case of weak F-SE arrays. We conclude this section with some notation and definitions. The random vector ðS; TÞ taking values in N2 is an adapted random time with respect to F if fS ¼ i; T ¼ jg 2 Fij ; 8i; jX1: It is a predictable random time if fS ¼ i; T ¼ jg 2 Fi1;j1 ; 8i; jX1: Similarly, the Zþ -valued random variable S is an F1 -adapted (predictable) random time if fS ¼ ig 2 F1i ðF1i1 Þ; 8iX1: F2 adapted and predictable random times are defined similarly. Note that if ðS; TÞ is F-adapted
ARTICLE IN PRESS B.G. Ivanoff, N.C. Weber / Statistics & Probability Letters 70 (2004) 95–108
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(predictable), then S and T are F1 - and F2 -adapted (predictable), respectively. However, in general the converse is not true.
3. Predictable sampling theorems 3.1. Strong F-SE arrays Our first result is a predictable sampling theorem for finite or infinite strong F-SE arrays. In this case, we have results involving both one and two dimensional predictable times. Theorem 3.1. Let X be a finite or infinite strong F-SE array. 1. Let S1 ; S 2 ; . . . ; Si be distinct, finite F1 -predictable times in f1; . . . ; cg and let T 1 ; T 2 ; . . . ; T j be distinct, finite F2 -predictable times in f1; . . . ; rg: Then ðX S1 ; . . . ; X Si Þ¼D ðX 1 ; . . . ; X i Þ and ðX T 1 ; . . . ; X T j Þ¼D ðX 1 ; . . . ; X j Þ: 2. Let S1 ; S 2 ; . . . ; Si be distinct, finite F1 -predictable times in f1; . . . ; cg and let T 1 ; T 2 ; . . . ; T j be distinct, finite F2 -predictable times in f1; . . . ; rg and suppose that one of the following two conditions is satisfied: (a) Sh is F20 -measurable, 8h ¼ 1; . . . ; i or T k is F10 -measurable, 8k ¼ 1; . . . ; j: (b) For all h ¼ 1; . . . ; i and k ¼ 1; . . . ; j; ðS h ; T k Þ are F-predictable times in f1; . . . ; cg f1; . . . ; rg: Then ðX Sh T k ; 1phpi; 1pkpjÞ¼D ðX hk ; 1phpi; 1pkpjÞ: Proof. 1. This is a consequence of Lemma 2.1 and Theorem 11.13 of Kallenberg (2002). 2(a) Assume that T k is F10 -measurable, 8k ¼ 1; . . . ; j: Let f hk : G ! R be bounded measurable functions, h ¼ 1; . . . ; i; k ¼ 1; . . . ; j: Then " !# " # j j i Y i Y Y Y E f hk ðX Sh T k Þ ¼ E E f hk ðX Sh T k Þ j F10 h¼1 k¼1
h¼1 k¼1
" ¼E E " ¼E " ¼E
j i Y Y h¼1 k¼1
j i Y Y h¼1 k¼1 j i Y Y h¼1 k¼1
!# f hk ðX hT k Þ j
F10
ð9Þ
#
f hk ðX hT k Þ # f hk ðX hk Þ :
ð10Þ
ARTICLE IN PRESS B.G. Ivanoff, N.C. Weber / Statistics & Probability Letters 70 (2004) 95–108
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We apply the predictable sampling theorem for exchangeable sequences (Kallenberg, 2002, Theorem 11.13) twice. Eq. (9) follows from the F1 -exchangeability of the column vectors and the fact that, given F10 ; the row permutations become fixed. Eq. (10) then follows from the F2 -exchangeability of the row vectors. Finish with a monotone class argument. 2(b) As in the proof of Theorem 11.13 of Kallenberg (2002), define the allocation sequences associated with ðS1 ; . . . ; S i Þ and ðT 1 ; . . . ; T j Þ; ah ¼ inff‘ : S‘ ¼ hg; 1phpi;
bk ¼ inff‘ : T ‘ ¼ kg; 1pkpj:
Next, we assume for the moment that i ¼ c and j ¼ r; so that ðS1 ; . . . ; S i Þ and ða1 ; . . . ; ai Þ are inverse random permutations of ð1; . . . ; cÞ; and ðT 1 ; . . . ; T j Þ and ðb1 ; . . . ; bj Þ are inverse random permutations of ð1; . . . ; rÞ: For each m 2 f0; . . . ; ig; put am h ¼ ah for all hpm; and define recursively m m am hþ1 ¼ minðf1; . . . ; ignfa1 ; . . . ; ah gÞ;
Similarly, for each n 2 f0; . . . ; jg; put
bnk
mphpi 1: ¼ bk for all kpn; and define recursively
bnkþ1
¼ minðf1; . . . ; jgnfbn1 ; . . . ; bnk gÞ; npkpj 1: n that ðam h ; bk Þ is Fm1;n1 measurable for 1phpi; and 1pkpj: Further whenever am1 ¼ ah ; and for kon; bnk ¼ bn1 ¼ bk : Let Dmn ¼ fðh; kÞ : hXm or kXng: k h
Note am h ¼ For any bounded measurable functions f hk on G, " # YY E f am bnk ðX hk Þ h
h
k
"
¼E E
YY h
Y Y "
"
f am bnk ðX hk ÞE
¼E
h
k
Y
Y
h
#! f am bnk ðX hk Þ j Fm1;n1 h
ðh;kÞ2Dmn
f am1 bn1 ðX hk ÞE
" Y
Y
k
h
hom kon
YY
f am bnk ðX hk Þ j Fm1;n1 h
hom kon
¼E
#!
k
Y Y
¼E
hom;
ðh;kÞ2Dmn
#
#! f am1 bn1 ðX hk Þ j Fm1;n1 h
k
f am1 bn1 ðX hk Þ : h
ð11Þ
k
; mphpig ¼ The conditional expectation in (11) follows from strong F-SE, noting that fam1 h n1 n ; mphpig and fb ; npkpjg ¼ fb ; npkpjg: fam k k h Without loss of generality assume that cpr: Now, ach ¼ ah ; brk ¼ bk ; a0h ¼ h; b0k ¼ k and exactly as in Kallenberg (2002), we may conclude that " E
YY h
k
# f hk ðX Sh ;T k Þ ¼ E
"
YY h
k
# f ah bk ðX hk Þ ¼ E
"
YY h
k
# f hbkrc ðX hk Þ :
(12)
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Next for 1pnpr; we use the conditional exchangeability of the row vectors to observe that, " # YY f hbnk ðX hk Þ E h
k
Y Y
¼E
h
f hbnk ðX hk ÞE
kon
" ¼E
h
f hbn1 ðX hk ÞE
k
#! f hbnk ðX hk Þ j F2n1
kXn
Y Y
k
kon
YY h
h
"
Y Y
¼E
" Y Y
h
#
kXn
#! f hbn1 ðX hk Þ j F2n1 k
f hbn1 ðX hk Þ : k
Thus from (12), recalling that b0k ¼ k; " # " # " # YY YY YY f hk ðX Sh ;T k Þ ¼ E f hbrc ðX hk Þ ¼ E f hk ðX hk Þ : E k h
k
h
k
h
k
A monotone class argument completes the proof when i ¼ c and j ¼ r: The extension to ioc and/or jor as well as infinite values of r and c is done exactly as in Kallenberg (2002). & 3.2. Weak F-SE arrays Next, we present a predictable sampling theorem for weak F-SE arrays. The statement is perhaps the least obvious of our results; the random times need to be totally ordered; that is, the relative orders of the sequence of first components must be identical to those of the second. Theorem 3.2. Let X be a finite or infinite weak c r F-SE array. If ðS1 ; T 1 Þ; . . . ; ðS n ; T n Þ is a sequence of finite F-predictable random times such that there exists a random permutation Z of ð1; . . . ; nÞ such that 1pS Zð1Þ oS Zð2Þ o oS ZðnÞ pc and 1pT Zð1Þ oT Zð2Þ o oT ZðnÞ pr; then ðX S1 T 1 ; . . . ; X Sn T n Þ¼D ðX 11 ; . . . ; X nn Þ: Proof. Because both sequences of coordinates have the same relative orders, the allocation sequence approach of Theorem 3.1 is applicable only to square arrays. Therefore, we shall use a less elegant but more direct and general argument.
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102
Let r denote a fixed (but arbitrary) permutation of ð1; . . . ; nÞ: As before, for 0pmpn we define the permutation rm as follows: rðhÞ; hpm; m r ðhÞ ¼ m m minðf1; . . . ; ngnfr ð1Þ; . . . ; r ðh 1ÞgÞ; h4m: Note that rm ðhÞ depends only on rð1Þ; . . . ; rðmÞ for all h and that r0 ðhÞ ¼ h: As well, for consistency in what follows, define rð0Þ ¼ 0: First assume c; ro1: For any bounded measurable functions f 1 ; . . . ; f n on G, " # n Y f i ðX Si ;T i Þ E i¼1
¼
X
X r
E
1parð1Þ ooarðnÞ pc 1pbrð1Þ oobrðnÞ pr
" n Y
f i ðX Si T i Þ
i¼1
¼
X r
n Y
X n Y
#
IðS i ¼ ai ; T i ¼ bi Þ
i¼1
X
X
E
1parð1Þ ooarðnÞ pc 1pbrð1Þ oobrðnÞ pr
¼
r
n Y
" n1 Y
f rðiÞ ðX arðiÞ brðiÞ Þ
i¼1
#
IðS rðiÞ ¼ arðiÞ ; T rðiÞ ¼ brðiÞ ÞEðf rðnÞ ðX arðnÞ brðnÞ ÞjFarðnÞ 1;brðnÞ 1 Þ
i¼1
X
ð13Þ
X
X
E
1parð1Þ ooarðnÞ pc 1pbrð1Þ oobrðnÞ pr
" n1 Y
ð14Þ
f rðiÞ ðX arðiÞ brðiÞ Þ
i¼1
#
IðS rðiÞ ¼ arðiÞ ; T rðiÞ ¼ brðiÞ ÞEðf rðnÞ ðX cr ÞjFarðnÞ 1;brðnÞ 1 Þ
ð15Þ
i¼1
since X is weak F-SE; ¼
X r
n1 Y i¼1
¼
X r
X
"
X
E f rðnÞ ðX cr Þ
1parð1Þ ooarðn1Þ pc1 1pbrð1Þ oobrðn1Þ pr1
IðS rðiÞ ¼ arðiÞ ; T rðiÞ ¼ brðiÞ ÞIðS rðnÞ 4arðn1Þ ; T rðnÞ 4brðn1Þ Þ X
h E f rðnÞ ðX cr Þ
X
1parð1Þ ooarðn1Þ pc1 1pbrð1Þ oobrðn1Þ pr1 n2 Y i¼1
i¼1
f rðiÞ ðX arðiÞ brðiÞ Þ
i¼1
#
Eðf rðn1Þ ðX arðn1Þ brðn1Þ ÞjFarðn1Þ 1;brðn1Þ 1 Þ n1 Y
n1 Y
f rðiÞ ðX arðiÞ brðiÞ Þ #
IðS rðiÞ ¼ arðiÞ ; T rðiÞ ¼ brðiÞ ÞIðS rðnÞ 4arðn1Þ ; T rðnÞ 4brðn1Þ Þ
ð16Þ
ARTICLE IN PRESS B.G. Ivanoff, N.C. Weber / Statistics & Probability Letters 70 (2004) 95–108
¼
X
X
h E f rðnÞ ðX cr Þf rðn1Þ ðX c1r1 Þ
X
1parð1Þ ooarðn1Þ pc1 1pbrð1Þ oobrðn1Þ pr1
r
103
n2 Y
f rðiÞ ðX arðiÞ brðiÞ Þ
i¼1
n1 Y
IðS rðiÞ ¼ arðiÞ ; T rðiÞ ¼ brðiÞ Þ
i¼1
IðS rðnÞ 4arðn1Þ ; T rðnÞ 4brðn1Þ Þ
ð17Þ
since X is weak F-SE; ¼
X r
X
"
X
1parð1Þ ooarðn1Þ pc1 1pbrð1Þ oobrðn1Þ pr1
E f rn2 ðnÞ ðX cr Þf rn2 ðn1Þ ðX c1r1 Þ
n2 Y
f rðiÞ ðX arðiÞ brðiÞ Þ
i¼1
n1 Y
IðS rðiÞ ¼ arðiÞ ; T rðiÞ ¼ brðiÞ ÞIðS rðnÞ 4arðn1Þ ; T rðnÞ 4brðn1Þ Þ
i¼1
¼
#
X
X
ð18Þ
h E f rn2 ðnÞ ðX cr Þ
X
rn2 1parn2 ð1Þ ooarn2 ðn2Þ pc2 1pbrn2 ð1Þ oobrn2 ðn2Þ pr2
f rn2 ðn1Þ ðX c1;r1 Þ
n2 Y
f rðiÞ ðX arðiÞ brðiÞ Þ
i¼1
n2 Y
IðS rðiÞ ¼ arðiÞ ; T rðiÞ ¼ brðiÞ Þ
i¼1
IðS rn2 ðn1Þ 4arðn2Þ ; T rn2 ðn1Þ 4brðn2Þ ÞIðS rn2 ðnÞ 4arðn2Þ ; T rn2 ðnÞ 4brðn2Þ Þ . ¼ .. X ¼
X
1park ð1Þ ooark ðkÞ pcðnkÞ 1pbrk ð1Þ oobrk ðkÞ prðnkÞ
rk
"
n Y
E
f rk ðjÞ ðX cðnjÞ;rðnjÞ Þ
k Y
IðS rðiÞ ¼ arðiÞ ; T rðiÞ ¼ brðiÞ Þ
i¼1
¼
f rðiÞ ðX arðiÞ brðiÞ Þ #
n Y
IðS rk ðhÞ 4arðkÞ ; T rk ðhÞ 4brðkÞ Þ
h¼kþ1
X
X
X
1park ð1Þ ooark ðkÞ pcðnkÞ 1pbrk ð1Þ oobrk ðkÞ prðnkÞ
rk
" E
k Y i¼1
j¼kþ1
X
n Y
j¼kþ1
f rk ðjÞ ðX cðnjÞ;rðnjÞ Þ
ð19Þ
ARTICLE IN PRESS B.G. Ivanoff, N.C. Weber / Statistics & Probability Letters 70 (2004) 95–108
104
k 1 Y
Eðf rðkÞ ðX arðkÞ brðkÞ ÞjFarðkÞ 1;brðkÞ 1 Þ
f rðiÞ ðX arðiÞ brðiÞ Þ
i¼1
k Y
IðS rðiÞ ¼ arðiÞ ; T rðiÞ ¼ brðiÞ Þ
i¼1
¼
E
X
1park ð1Þ ooark ðkÞ pcðnkÞ 1pbrk ð1Þ oobrk ðkÞ prðnkÞ
" n Y
f rk1 ðjÞ ðX cðnjÞ;rðnjÞ Þ
k Y
k1 Y
f rðiÞ ðX arðiÞ brðiÞ Þ
i¼1
IðS rðiÞ ¼ arðiÞ ; T rðiÞ ¼ brðiÞ Þ
i¼1
¼
ð20Þ
X
j¼k
IðSrk ðhÞ 4arðkÞ ; T rk ðhÞ 4brðkÞ Þ
h¼kþ1
X rk
#
n Y
X
n Y
# IðS rk ðhÞ 4arðkÞ ; T rk ðhÞ 4brðkÞ Þ
h¼kþ1
X
ð21Þ
X
rk1 1park1 ð1Þ ooark1 ðk1Þ pcðnkþ1Þ 1pbrk1 ð1Þ oobrk1 ðk1Þ prðnkþ1Þ
E
" n Y
f rk1 ðjÞ ðX cðnjÞ;rðnjÞ Þ
IðSrðiÞ ¼ arðiÞ ; T rðiÞ ¼ brðiÞ Þ
i¼1
. ¼ ..
"
¼E " ¼E " ¼E
n Y
j¼1 n Y
n Y
# IðSrk1 ðhÞ 4arðk1Þ ; T rk1 ðhÞ 4brðk1Þ Þ
ð22Þ
h¼k
f r0 ðjÞ ðX cðnjÞ;rðnjÞ Þ
j¼1 n Y
f rðiÞ ðX arðiÞ brðiÞ Þ
i¼1
j¼k k1 Y
k1 Y
#
n Y
# IðSr0 ðhÞ 4arð0Þ ; T r0 ðhÞ 4brð0Þ Þ
h¼1
f j ðX cðnjÞ;rðnjÞ Þ # f j ðX jj Þ ;
by exchangeability.
j¼1
P Note that r denotes summation over all possible n! permutations of ð1; . . . ; nÞ: The predictability of ðSrðnÞ ; T rðnÞ Þ and the total order of ðarðiÞ ; brðiÞ Þ are used to obtain (14). Total ordering is again used to enable us to sum over arðnÞ ; brðnÞ to obtain (16). Eq. (18) follows by using the weak F-SE property after conditioning on Fc2;r2 : Since arðn1Þ1 pc 2 and brðn1Þ1 pr 2; the indicator functions in (18) are all Fc2;r2 measurable. Since IðSrn2 ðnÞ ; Srn2 ðn1Þ 4arðn2Þ ; T rn2 ðnÞ ; T rn2 ðn1Þ 4brðn2Þ Þ is Farðn2Þ 1;brðn2Þ 1 measurable Eq. (19) follows after summing over arðn1Þ ; brðn1Þ and the last two terms of r: Eq. (20) follows by successive conditioning on FarðkÞ1 ;brðkÞ1 : Finally Eq. (22) follows as in (19).
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An extension to infinite arrays and unbounded predictable times follows as in the proof of Theorem 11.3 in Kallenberg (2002). & 3.3. Strong F-JE arrays The following result is the analogue of Theorem 4.1.2 in Ivanoff and Weber (2004), a predictable sampling theorem for finite and infinite strong F-JE arrays. Theorem 3.3. Let X be a finite or infinite strong n n F-JE array and suppose that t1 ; . . . ; tk are distinct, finite Fii -predictable times. Then ðX ti tj ; 1pi; jpkÞ¼D ðX ij ; 1pi; jpkÞ: Proof. We begin by assuming that no1: Since the array is square, we may proceed by using allocation sequences. Define ai ¼ inff‘; t‘ ¼ ig;
1pipn:
Initially assume that n ¼ k; so that ðt1 ; . . . ; tk Þ and ða1 ; . . . ; ak Þ are inverse random permutations of ð1; . . . ; nÞ: For each m 2 f0; . . . ; kg; put am i ¼ ai for all ipm; and define recursively m m am iþ1 ¼ minðf1; . . . ; kgnfa1 ; . . . ; ai gÞ;
mpipk 1:
m ðam 1 ; . . . ; ak Þ
m1 We have that is Fm1;m1 measurable, and whenever iom; am ¼ ai : Recall i ¼ ai Dmm ¼ fði; jÞ : iXm or jXmg: For any bounded measurable functions f hk on G, " # YY E f ami amj ðX ij Þ i
j
Y Y
¼E
iom jom
Y Y
¼E " ¼E
iom jom
YY i
j
" f ami amj ðX ij ÞE
Y
#!
Y
f ami amj ðX ij Þ j Fm1;m1
ði;jÞ2Dmm
f am1 ðX ij ÞE am1 i j
" Y
Y
#! f am1 ðX ij Þ j Fm1;m1 am1 i j
ð23Þ
ði;jÞ2Dmm
#
f am1 ajm1 ðX ij Þ : i
The conditional expectation in (23) follows from strong F-JE, noting that fam1 ; mpipkg ¼ i m k 0 fai ; mpipkg: Now, ai ¼ ai ; ai ¼ i; and as in Kallenberg (2002), we may conclude that " # " # " # YY YY YY E f ij ðX ti tj Þ ¼ E f ai aj ðX ij Þ ¼ E f ij ðX ij Þ : i
j
i
j
i
j
A monotone class argument completes the proof when k ¼ n: The extension to general values of k and infinite values of n is done as in Kallenberg (2002). &
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3.4. Weak F-JE arrays For completeness we include the following predictable sampling result for weak F-JE arrays which is an immediate consequence of the diagonal terms being conditionally exchangeable given fFii g: Lemma 3.4. Let X be a finite or infinite weak F-JE array and suppose that t1 ; . . . ; tk are distinct, finite Fii -predictable times. Then ðX t1 t1 ; . . . ; X tk tk Þ¼D ðX 11 ; . . . ; X kk Þ:
4. Infinite and Ergodic arrays We now consider situations in which exchangeability is equivalent to the weaker concept of spreadability. We recall that an array is separately (respectively, jointly) spreadable ((SS, respectively JS) if (3) (respectively, (5)) is satisfied for increasing subsequences of indices. The weak and strong F-SS and JS properties are analogous to the corresponding properties for exchangeable arrays (for details, see Ivanoff and Weber (2004)). 4.1. Infinite arrays For infinite sequences, it is well known that conditional spreadability and exchangeability are equivalent. Therefore, it is an immediate consequence of Lemma 2.1 that if X is an infinite array, X is strong F-separately spreadable (F-SS) if and only if X is strong F-SE. We continue with an infinite weak F-SS array X. We say that X is F-stationary if yST X ¼D X for every bounded adapted random time ðS; TÞ: By Theorem 1 of Ivanoff and Weber (1996), X is weak F-SE if and only if X is F-stationary so in this case, by Lemma 3.2.1 of Ivanoff and Weber (2004) the weak F-SE property is equivalent to the weak F-SS property. In the case of the joint properties, even in the case of infinite arrays joint spreadability is not strictly equivalent to joint exchangeability. If X is JE, then it is straightforward by simply permuting i and j that X ij ¼D X ji : Thus, all the diagonal elements are identically distributed as are all the off-diagonal elements, although the two distributions may be different. In the case of a JS array, the diagonal elements have the same distribution, as do the random variables ðX ij ; iojÞ and the random variables ðX ij ; i4jÞ; but the three distributions may be different. However, Kallenberg (1992) notes that in the infinite case an array is JS if and only if the array X ‘‘folded’’ on the diagonal (i.e. the upper triangular array Y :¼ðY ij ¼ ðX ij ; X ji Þ; ipjÞ) can be embedded in a JE array. 4.2. Finite Ergodic arrays An ergodic array is generated by random permutations of the rows and columns of a finite array of constants. In this situation we shall see that under certain conditions, spreadability and
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exchangeability are equivalent, as are the weak and strong F properties. First we consider the SE and SS properties. Lemma 4.1. Let X be an array generated by independent random permutations p and s of the columns and rows of a (finite) c r array of distinct constants and let Fij ¼ sfpð1Þ; . . . ; pðiÞ; sð1Þ; . . . ; sðjÞg: Then the following are equivalent: 1. 2. 3. 4.
X ST ¼D X 11 for every F-predictable random time ðS; TÞ: X is weak F-SS. X is weak F-SE. X is strong F-SE.
Proof. We have that 4 ) 3 ) 2 and it was shown in Theorem 3.2.3 in Ivanoff and Weber (2004) that 2 ) 1: It remains to prove 1 ) 4: It was proven in Theorem 4 of Ivanoff and Weber (1996) that 133: If X ST ¼D X 11 for every F-predictable random time ðS; TÞ; then examining their proof, it is seen that given Fij ; the remaining column and row permutations are independently and uniformly distributed over the ðc iÞ! and ðr jÞ! possibilities, respectively. This implies the appropriate conditional distribution for the entire shell outside ðX hk ; 1phpi; 1pkpjÞ; and so it follows that X is strong F-SE. & We move now to the joint exchangeability and spreadability properties. Lemma 4.2. Let X be a finite n n array generated by applying the same random permutation to the rows and columns of an array of constants where the diagonal of the array consists of n distinct elements and the diagonal and off-diagonal entries form disjoint sets. The following are equivalent: 1. 2. 3. 4.
X SS ¼D X 11 for every Fii -predictable random time S. X is weak F-JS. X is weak F-JE. X is strong F-JE.
Proof. The key point in the JE case is that the random permutation, p; is determined by the behaviour of the diagonal elements, and so the proof is essentially one dimensional. The result then follows from Lemma 3.4 and by arguing as in the proof of Lemma 4.1. &
References Aldous, D.J., 1985. Exchangeability and related topics. In: E´cole d’e´te´ de probabilitie´s de Saint-Flour XIII. Lecture Notes in Mathematics, P.L. Hennequin (Ed.), vol. 1117. Springer, Berlin. De Finetti, B., 1929. Fuzione caratteristica di un fenomeno aleatorio. Atti Congr. Int. Mat. Bologna 1928 6, 179–190. De Finetti, B., 1930. Fuzione caratteristica di un fenomeno aleatorio. Mem. Roy. Acc. Lincei 4, 86–133. Ivanoff, B.G., Weber, N.C., 1996. Some characterizations of partial exchangeability. J. Austral. Math. Soc. (A) 61, 345–359. Ivanoff, B.G., Weber, N.C., 2004. Spreadable arrays and martingale structures. J. Austral. Math. Soc., to appear. Kallenberg, O., 1982. Characterizations and embedding properties in exchangeability. Z. Wahrsch. Verw. Gebiete 60, 249–281.
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Kallenberg, O., 1988. Spreading and predictable sampling in exchangeable sequences and processes. Ann. Probab. 16, 508–534. Kallenberg, O., 1992. Symmetries on random arrays and set-indexed processes. J. Theoret. Probab. 5, 727–765. Kallenberg, O., 2000. Spreading-invariant sequences and processes on bounded index sets. Probab. Theory Related Fields 118, 211–250. Kallenberg, O., 2002. Foundations of Modern Probability, second ed. Springer, New York. Kingman, J.F.C., 1978. Uses of exchangeability. Ann. Probab. 6, 183–197. Ryll-Nardzewski, C., 1957. On stationary sequences of random variables and the de Finetti’s equivalence. Colloq. Math. 4, 149–156.
Statistics & Probability Letters 70 (2004) 95–108 www.elsevier.com/locate/stapro
Predictable sampling for partially exchangeable arrays B. Gail Ivanoffa,1, N.C. Weberb, a
Department of Mathematics and Statistics, University of Ottawa, P.O. Box 450 Station A, Ottawa, Ont., Canada K1N 6N5 b School of Mathematics and Statistics, F07, University of Sydney, Sydney, NSW 2006, Australia Received 16 March 2004; received in revised form 15 June 2004
Abstract New predictable sampling theorems are developed for two natural extensions of the concept of F-exchangeability applied to separately exchangeable arrays and to jointly exchangeable arrays. Further we show that weak and strong F-exchangeability are in fact the same for certain ergodic arrays. r 2004 Elsevier B.V. All rights reserved. MSC: primary 60G09 Keywords: Separately exchangeable array; Jointly exchangeable array; Adapted time; Spreadable array; Predictable sampling
1. Introduction Exchangeable sequences and arrays of random elements have been studied extensively since the seminal work of De Finetti (1929, 1930). A finite or infinite sequence is said to be exchangeable if its distribution is invariant under permutations that leave all but a finite number of indices fixed. There is an extensive literature on exchangeable and related arrays. Aldous (1985) and Kallenberg (2002) provide excellent general coverage of the theory of exchangeability. Corresponding author. 1
E-mail addresses: [email protected] (B.G. Ivanoff), [email protected] (N.C. Weber). Research supported by a grant from the Natural Sciences and Engineering Research Council of Canada.
0167-7152/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2004.08.005
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Kallenberg (1982) introduced the concept of conditional exchangeability for sequences by relating exchangeability to a filtration F to which the sequence is adapted, and in Kallenberg (1988) proved a predictable sampling theorem: i.e. that the joint distribution of the first k elements of the sequence is the same as that of the sequence sampled at k distinct, F-predictable random times. Exchangeability is in general a stronger concept than spreadability: a sequence is spreadable if all finite subsequences of the same size have the same distribution. Kallenberg (2000) proves a weaker version of the predictable sampling theorem for conditionally spreadable sequences which requires the F-predictable random times to be strictly increasing. Nonetheless, there are situations in which spreadability is equivalent to exchangeability: in particular, if the sequence is infinite, see Ryll-Nardzewski (1957), or finite and ergodic, see Kallenberg (2002). Extending the ideas of exchangeability and spreadability to two-dimensional arrays leads to two different approaches. For the stronger separate exchangeability the joint distribution of the array is invariant under permutations of both rows and columns. If the joint distribution of the array is invariant under the same permutation of rows and columns then we say the array is jointly exchangeable. Separate and joint spreadability are defined analogously (see, Kallenberg, 1992; Ivanoff and Weber, 1996, 2004). The extension of conditional (F-) exchangeability (respectively, spreadability) to arrays is not entirely straightforward. In Ivanoff and Weber (1996) F-exchangeability of arrays was defined in two different ways and extensions of F-spreadability to arrays was explored in Ivanoff and Weber (2004) where the relationship between the different types of F-spreadability and twodimensional martingale structures was investigated. Predictable sampling theorems were proven for the various F-spreadable structures in that paper. The primary purpose of this note is to develop predictable sampling theorems for each of the four types of conditionally exchangeable arrays (see Section 3). In light of the corresponding theorems in Kallenberg (2000) and Ivanoff and Weber (2004), the appropriate statement of the relevant theorems can be anticipated; in some cases, the proof is a reasonably straightforward adaptation of one-dimensional techniques. However, in at least one case, a new and distinctly two-dimensional approach is required. We also explore conditions under which a spreadable array is exchangeable in Section 4. We will show that certain results for ergodic arrays in Ivanoff and Weber (1996) can be strengthened and that in these cases spreadability and exchangeability are equivalent.
2. Notation and definitions A finite or infinite random sequence x ¼ ðx1 ; x2 ; . . .Þ in a measurable space ðG; GÞ is said to be exchangeable if ðxk1 ; . . . ; xkn Þ¼D ðx1 ; . . . ; xn Þ
(1)
for any collection k1 ; . . . ; kn of distinct elements in the index set of x: Equivalently, X is exchangeable if for any permutation p of ð1; 2; . . .Þ leaving all but a finite number of terms
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unchanged, X p ¼ ðX pð1Þ ; X pð2Þ ; . . .Þ¼D ðX 1 ; X 2 ; . . .Þ ¼ X :
(2)
We say that x is spreadable if (1) holds whenever k1 o okn ; i.e. all subsequences of the same size have the same distribution. In the infinite case Ryll-Nardzewski (1957) showed that spreadability and exchangeability are in fact equivalent but as Kingman (1978) noted this is not the case for finite sequences. For clarity and notational convenience in what follows, given a c r array we shall generally assume that either both c; ro1 or c ¼ r ¼ 1: However, all definitions and theorems can be adapted in a straightforward way to include the case in which only one of c and r is finite. Let X ¼ ðX ij : 1pipc; 1pjprÞ be a c r array of random elements taking their values in some measurable space ðG; GÞ: X is separately exchangeable (SE) if for all moc; qor; m; qo1 ðX ik j h : 1pkpm; 1phpqÞ¼D ðX ij : 1pipm; 1pjpqÞ
(3)
holds for distinct, but not necessarily increasing, values i1 ; . . . ; im and j 1 ; . . . ; j q : (The array is separately spreadable (SS) if (3) is satisfied for all strictly increasing values 1pi1 o oim pc and 1pj 1 o oj q pr:) Alternatively, X is SE if and only if for any permutations p of ð1; . . . ; cÞ and s of ð1; . . . ; rÞ each leaving all but finitely many terms unchanged, X p ¼D X
and X s ¼D X :
(4)
Here, X p ¼ ðX pðiÞj : 1pipc; 1pjprÞ and X s ¼ ðX isðjÞ : 1pipc; 1pjprÞ: We have X ps :¼ðX p Þs ¼ ðX s Þp : The square array X ¼ ðX ij : 1pi; jpnÞ (n may be 1) is jointly exchangeable (JE) if for all finite mon ðX ik ih : 1pk; hpmÞ¼D ðX ij : 1pi; jpmÞ
(5)
holds for distinct, but not necessarily increasing, values i1 ; . . . ; im : (The array is jointly spreadable (JS) is (5) is satisfied for all strictly increasing values 1pi1 o oim pn:) X is JE if and only if for any permutation p of ð1; . . . ; nÞ leaving all but finitely many terms unchanged, X pp ¼D X :
(6)
In order to define conditional exchangeability for arrays, we assume that we are given an arbitrary filtration F ¼ ðFij : 0pipc; 0pjprÞ to which X is adapted (i.e. Fij Fhk if iph and jpk; and X ij is Fij -measurable, 8i; j). Roughly speaking, we will say that an array has a particular property in the weak F sense if, conditioned on Fjk ; the translated array yjk X :¼ ðX h‘ : j þ 1phpc; k þ 1p‘prÞ has the required property for all ðj; kÞ 2 f1; . . . ; cg f1; . . . ; rg: On the other hand, assuming that all necessary conditional distributions exist, suppose that the actions defining the property in question are carried out in such a way that the first j columns and k rows are left unchanged and consider the conditional distribution of the entire array given Fjk : Then we say that the array has the property in the strong F sense if the conditional distributions of the new array and the original are the same for all ðj; kÞ 2 f1; . . . ; cg f1; . . . ; rg:
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More precisely, the array c r X is weak F-SE if conditioned on Fjk ; yjk X is SE;
8joc and 8kor:
(7)
Similarly, the n n array X is weak F-JE if conditioned on Fkk ; ykk X is JE;
8kon:
(8)
The array X is strong F-SE if for all ipc and all jpr and any finite permutations p of ð1; . . . ; cÞ leaving ð1; . . . ; iÞ unchanged and s of ð1; . . . ; rÞ leaving ð1; . . . ; jÞ unchanged, X p ¼DjFij X
and X s ¼DjFij X ;
where ‘¼DjFij ’ denotes equality of the conditional distributions given Fij : The n n array X is strong F-JE if, for every ipc and any finite permutation p of ð1; . . . ; nÞ leaving ð1; . . . ; iÞ unchanged, X pp ¼DjFii X : The concept of strong F-exchangeability was introduced only for finite arrays in Ivanoff and Weber (1996); here we consider it for both finite and infinite arrays, and will see that in some sense it is the more appropriate analogue of the one-dimensional conditioning. It is clear that the strong F-properties imply the weak F-properties, and they are equivalent when F is the minimal filtration generated by the array. For general filtrations, the converse is not true: an example is given in Ivanoff and Weber (1996, p. 350) that is weak F-SE but not strong F-SE. Given the filtration F ¼ ðFij : 0pipc; 0pjprÞ denote the (one-dimensional) filtrations F1 ¼ ðF1i : 1pipcÞ and F2 ¼ ðF2j : 1pjprÞ; where F1i ¼ _rj¼1 Fij and F2j ¼ _ci¼1 Fij : Arguing as in Lemma 3.1.1 of Ivanoff and Weber (2004) we have the following result. Lemma 2.1. Let X be a finite or infinite array. X is strong F-SE if and only if the sequence of column vectors ðX i ; 1pipcÞ is F1 -exchangeable and the sequence of row vectors ðX j ; 1pjprÞ is F2 -exchangeable. Lemma 2.1 improves on Lemmas 2 and 3 of Ivanoff and Weber (1996) in a number of ways. First, conditional independence of F1i and F2j given Fij was assumed, but this condition is not needed. Second, the conclusion of Lemma 2 was that F1 - and F2 -exchangeability of the column and row vectors of an infinite array implies that X is weak F-SE; we see here that in fact X is strong F-SE. This stronger result was proven in Lemma 3 of Ivanoff and Weber (1996) in the case of finite arrays, but it was incorrectly stated subsequently that this is not a necessary condition. In fact, F1 - and F2 -exchangeability of the column and row vectors is not a necessary condition only in the case of weak F-SE arrays. We conclude this section with some notation and definitions. The random vector ðS; TÞ taking values in N2 is an adapted random time with respect to F if fS ¼ i; T ¼ jg 2 Fij ; 8i; jX1: It is a predictable random time if fS ¼ i; T ¼ jg 2 Fi1;j1 ; 8i; jX1: Similarly, the Zþ -valued random variable S is an F1 -adapted (predictable) random time if fS ¼ ig 2 F1i ðF1i1 Þ; 8iX1: F2 adapted and predictable random times are defined similarly. Note that if ðS; TÞ is F-adapted
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(predictable), then S and T are F1 - and F2 -adapted (predictable), respectively. However, in general the converse is not true.
3. Predictable sampling theorems 3.1. Strong F-SE arrays Our first result is a predictable sampling theorem for finite or infinite strong F-SE arrays. In this case, we have results involving both one and two dimensional predictable times. Theorem 3.1. Let X be a finite or infinite strong F-SE array. 1. Let S1 ; S 2 ; . . . ; Si be distinct, finite F1 -predictable times in f1; . . . ; cg and let T 1 ; T 2 ; . . . ; T j be distinct, finite F2 -predictable times in f1; . . . ; rg: Then ðX S1 ; . . . ; X Si Þ¼D ðX 1 ; . . . ; X i Þ and ðX T 1 ; . . . ; X T j Þ¼D ðX 1 ; . . . ; X j Þ: 2. Let S1 ; S 2 ; . . . ; Si be distinct, finite F1 -predictable times in f1; . . . ; cg and let T 1 ; T 2 ; . . . ; T j be distinct, finite F2 -predictable times in f1; . . . ; rg and suppose that one of the following two conditions is satisfied: (a) Sh is F20 -measurable, 8h ¼ 1; . . . ; i or T k is F10 -measurable, 8k ¼ 1; . . . ; j: (b) For all h ¼ 1; . . . ; i and k ¼ 1; . . . ; j; ðS h ; T k Þ are F-predictable times in f1; . . . ; cg f1; . . . ; rg: Then ðX Sh T k ; 1phpi; 1pkpjÞ¼D ðX hk ; 1phpi; 1pkpjÞ: Proof. 1. This is a consequence of Lemma 2.1 and Theorem 11.13 of Kallenberg (2002). 2(a) Assume that T k is F10 -measurable, 8k ¼ 1; . . . ; j: Let f hk : G ! R be bounded measurable functions, h ¼ 1; . . . ; i; k ¼ 1; . . . ; j: Then " !# " # j j i Y i Y Y Y E f hk ðX Sh T k Þ ¼ E E f hk ðX Sh T k Þ j F10 h¼1 k¼1
h¼1 k¼1
" ¼E E " ¼E " ¼E
j i Y Y h¼1 k¼1
j i Y Y h¼1 k¼1 j i Y Y h¼1 k¼1
!# f hk ðX hT k Þ j
F10
ð9Þ
#
f hk ðX hT k Þ # f hk ðX hk Þ :
ð10Þ
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We apply the predictable sampling theorem for exchangeable sequences (Kallenberg, 2002, Theorem 11.13) twice. Eq. (9) follows from the F1 -exchangeability of the column vectors and the fact that, given F10 ; the row permutations become fixed. Eq. (10) then follows from the F2 -exchangeability of the row vectors. Finish with a monotone class argument. 2(b) As in the proof of Theorem 11.13 of Kallenberg (2002), define the allocation sequences associated with ðS1 ; . . . ; S i Þ and ðT 1 ; . . . ; T j Þ; ah ¼ inff‘ : S‘ ¼ hg; 1phpi;
bk ¼ inff‘ : T ‘ ¼ kg; 1pkpj:
Next, we assume for the moment that i ¼ c and j ¼ r; so that ðS1 ; . . . ; S i Þ and ða1 ; . . . ; ai Þ are inverse random permutations of ð1; . . . ; cÞ; and ðT 1 ; . . . ; T j Þ and ðb1 ; . . . ; bj Þ are inverse random permutations of ð1; . . . ; rÞ: For each m 2 f0; . . . ; ig; put am h ¼ ah for all hpm; and define recursively m m am hþ1 ¼ minðf1; . . . ; ignfa1 ; . . . ; ah gÞ;
Similarly, for each n 2 f0; . . . ; jg; put
bnk
mphpi 1: ¼ bk for all kpn; and define recursively
bnkþ1
¼ minðf1; . . . ; jgnfbn1 ; . . . ; bnk gÞ; npkpj 1: n that ðam h ; bk Þ is Fm1;n1 measurable for 1phpi; and 1pkpj: Further whenever am1 ¼ ah ; and for kon; bnk ¼ bn1 ¼ bk : Let Dmn ¼ fðh; kÞ : hXm or kXng: k h
Note am h ¼ For any bounded measurable functions f hk on G, " # YY E f am bnk ðX hk Þ h
h
k
"
¼E E
YY h
Y Y "
"
f am bnk ðX hk ÞE
¼E
h
k
Y
Y
h
#! f am bnk ðX hk Þ j Fm1;n1 h
ðh;kÞ2Dmn
f am1 bn1 ðX hk ÞE
" Y
Y
k
h
hom kon
YY
f am bnk ðX hk Þ j Fm1;n1 h
hom kon
¼E
#!
k
Y Y
¼E
hom;
ðh;kÞ2Dmn
#
#! f am1 bn1 ðX hk Þ j Fm1;n1 h
k
f am1 bn1 ðX hk Þ : h
ð11Þ
k
; mphpig ¼ The conditional expectation in (11) follows from strong F-SE, noting that fam1 h n1 n ; mphpig and fb ; npkpjg ¼ fb ; npkpjg: fam k k h Without loss of generality assume that cpr: Now, ach ¼ ah ; brk ¼ bk ; a0h ¼ h; b0k ¼ k and exactly as in Kallenberg (2002), we may conclude that " E
YY h
k
# f hk ðX Sh ;T k Þ ¼ E
"
YY h
k
# f ah bk ðX hk Þ ¼ E
"
YY h
k
# f hbkrc ðX hk Þ :
(12)
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Next for 1pnpr; we use the conditional exchangeability of the row vectors to observe that, " # YY f hbnk ðX hk Þ E h
k
Y Y
¼E
h
f hbnk ðX hk ÞE
kon
" ¼E
h
f hbn1 ðX hk ÞE
k
#! f hbnk ðX hk Þ j F2n1
kXn
Y Y
k
kon
YY h
h
"
Y Y
¼E
" Y Y
h
#
kXn
#! f hbn1 ðX hk Þ j F2n1 k
f hbn1 ðX hk Þ : k
Thus from (12), recalling that b0k ¼ k; " # " # " # YY YY YY f hk ðX Sh ;T k Þ ¼ E f hbrc ðX hk Þ ¼ E f hk ðX hk Þ : E k h
k
h
k
h
k
A monotone class argument completes the proof when i ¼ c and j ¼ r: The extension to ioc and/or jor as well as infinite values of r and c is done exactly as in Kallenberg (2002). & 3.2. Weak F-SE arrays Next, we present a predictable sampling theorem for weak F-SE arrays. The statement is perhaps the least obvious of our results; the random times need to be totally ordered; that is, the relative orders of the sequence of first components must be identical to those of the second. Theorem 3.2. Let X be a finite or infinite weak c r F-SE array. If ðS1 ; T 1 Þ; . . . ; ðS n ; T n Þ is a sequence of finite F-predictable random times such that there exists a random permutation Z of ð1; . . . ; nÞ such that 1pS Zð1Þ oS Zð2Þ o oS ZðnÞ pc and 1pT Zð1Þ oT Zð2Þ o oT ZðnÞ pr; then ðX S1 T 1 ; . . . ; X Sn T n Þ¼D ðX 11 ; . . . ; X nn Þ: Proof. Because both sequences of coordinates have the same relative orders, the allocation sequence approach of Theorem 3.1 is applicable only to square arrays. Therefore, we shall use a less elegant but more direct and general argument.
ARTICLE IN PRESS B.G. Ivanoff, N.C. Weber / Statistics & Probability Letters 70 (2004) 95–108
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Let r denote a fixed (but arbitrary) permutation of ð1; . . . ; nÞ: As before, for 0pmpn we define the permutation rm as follows: rðhÞ; hpm; m r ðhÞ ¼ m m minðf1; . . . ; ngnfr ð1Þ; . . . ; r ðh 1ÞgÞ; h4m: Note that rm ðhÞ depends only on rð1Þ; . . . ; rðmÞ for all h and that r0 ðhÞ ¼ h: As well, for consistency in what follows, define rð0Þ ¼ 0: First assume c; ro1: For any bounded measurable functions f 1 ; . . . ; f n on G, " # n Y f i ðX Si ;T i Þ E i¼1
¼
X
X r
E
1parð1Þ ooarðnÞ pc 1pbrð1Þ oobrðnÞ pr
" n Y
f i ðX Si T i Þ
i¼1
¼
X r
n Y
X n Y
#
IðS i ¼ ai ; T i ¼ bi Þ
i¼1
X
X
E
1parð1Þ ooarðnÞ pc 1pbrð1Þ oobrðnÞ pr
¼
r
n Y
" n1 Y
f rðiÞ ðX arðiÞ brðiÞ Þ
i¼1
#
IðS rðiÞ ¼ arðiÞ ; T rðiÞ ¼ brðiÞ ÞEðf rðnÞ ðX arðnÞ brðnÞ ÞjFarðnÞ 1;brðnÞ 1 Þ
i¼1
X
ð13Þ
X
X
E
1parð1Þ ooarðnÞ pc 1pbrð1Þ oobrðnÞ pr
" n1 Y
ð14Þ
f rðiÞ ðX arðiÞ brðiÞ Þ
i¼1
#
IðS rðiÞ ¼ arðiÞ ; T rðiÞ ¼ brðiÞ ÞEðf rðnÞ ðX cr ÞjFarðnÞ 1;brðnÞ 1 Þ
ð15Þ
i¼1
since X is weak F-SE; ¼
X r
n1 Y i¼1
¼
X r
X
"
X
E f rðnÞ ðX cr Þ
1parð1Þ ooarðn1Þ pc1 1pbrð1Þ oobrðn1Þ pr1
IðS rðiÞ ¼ arðiÞ ; T rðiÞ ¼ brðiÞ ÞIðS rðnÞ 4arðn1Þ ; T rðnÞ 4brðn1Þ Þ X
h E f rðnÞ ðX cr Þ
X
1parð1Þ ooarðn1Þ pc1 1pbrð1Þ oobrðn1Þ pr1 n2 Y i¼1
i¼1
f rðiÞ ðX arðiÞ brðiÞ Þ
i¼1
#
Eðf rðn1Þ ðX arðn1Þ brðn1Þ ÞjFarðn1Þ 1;brðn1Þ 1 Þ n1 Y
n1 Y
f rðiÞ ðX arðiÞ brðiÞ Þ #
IðS rðiÞ ¼ arðiÞ ; T rðiÞ ¼ brðiÞ ÞIðS rðnÞ 4arðn1Þ ; T rðnÞ 4brðn1Þ Þ
ð16Þ
ARTICLE IN PRESS B.G. Ivanoff, N.C. Weber / Statistics & Probability Letters 70 (2004) 95–108
¼
X
X
h E f rðnÞ ðX cr Þf rðn1Þ ðX c1r1 Þ
X
1parð1Þ ooarðn1Þ pc1 1pbrð1Þ oobrðn1Þ pr1
r
103
n2 Y
f rðiÞ ðX arðiÞ brðiÞ Þ
i¼1
n1 Y
IðS rðiÞ ¼ arðiÞ ; T rðiÞ ¼ brðiÞ Þ
i¼1
IðS rðnÞ 4arðn1Þ ; T rðnÞ 4brðn1Þ Þ
ð17Þ
since X is weak F-SE; ¼
X r
X
"
X
1parð1Þ ooarðn1Þ pc1 1pbrð1Þ oobrðn1Þ pr1
E f rn2 ðnÞ ðX cr Þf rn2 ðn1Þ ðX c1r1 Þ
n2 Y
f rðiÞ ðX arðiÞ brðiÞ Þ
i¼1
n1 Y
IðS rðiÞ ¼ arðiÞ ; T rðiÞ ¼ brðiÞ ÞIðS rðnÞ 4arðn1Þ ; T rðnÞ 4brðn1Þ Þ
i¼1
¼
#
X
X
ð18Þ
h E f rn2 ðnÞ ðX cr Þ
X
rn2 1parn2 ð1Þ ooarn2 ðn2Þ pc2 1pbrn2 ð1Þ oobrn2 ðn2Þ pr2
f rn2 ðn1Þ ðX c1;r1 Þ
n2 Y
f rðiÞ ðX arðiÞ brðiÞ Þ
i¼1
n2 Y
IðS rðiÞ ¼ arðiÞ ; T rðiÞ ¼ brðiÞ Þ
i¼1
IðS rn2 ðn1Þ 4arðn2Þ ; T rn2 ðn1Þ 4brðn2Þ ÞIðS rn2 ðnÞ 4arðn2Þ ; T rn2 ðnÞ 4brðn2Þ Þ . ¼ .. X ¼
X
1park ð1Þ ooark ðkÞ pcðnkÞ 1pbrk ð1Þ oobrk ðkÞ prðnkÞ
rk
"
n Y
E
f rk ðjÞ ðX cðnjÞ;rðnjÞ Þ
k Y
IðS rðiÞ ¼ arðiÞ ; T rðiÞ ¼ brðiÞ Þ
i¼1
¼
f rðiÞ ðX arðiÞ brðiÞ Þ #
n Y
IðS rk ðhÞ 4arðkÞ ; T rk ðhÞ 4brðkÞ Þ
h¼kþ1
X
X
X
1park ð1Þ ooark ðkÞ pcðnkÞ 1pbrk ð1Þ oobrk ðkÞ prðnkÞ
rk
" E
k Y i¼1
j¼kþ1
X
n Y
j¼kþ1
f rk ðjÞ ðX cðnjÞ;rðnjÞ Þ
ð19Þ
ARTICLE IN PRESS B.G. Ivanoff, N.C. Weber / Statistics & Probability Letters 70 (2004) 95–108
104
k 1 Y
Eðf rðkÞ ðX arðkÞ brðkÞ ÞjFarðkÞ 1;brðkÞ 1 Þ
f rðiÞ ðX arðiÞ brðiÞ Þ
i¼1
k Y
IðS rðiÞ ¼ arðiÞ ; T rðiÞ ¼ brðiÞ Þ
i¼1
¼
E
X
1park ð1Þ ooark ðkÞ pcðnkÞ 1pbrk ð1Þ oobrk ðkÞ prðnkÞ
" n Y
f rk1 ðjÞ ðX cðnjÞ;rðnjÞ Þ
k Y
k1 Y
f rðiÞ ðX arðiÞ brðiÞ Þ
i¼1
IðS rðiÞ ¼ arðiÞ ; T rðiÞ ¼ brðiÞ Þ
i¼1
¼
ð20Þ
X
j¼k
IðSrk ðhÞ 4arðkÞ ; T rk ðhÞ 4brðkÞ Þ
h¼kþ1
X rk
#
n Y
X
n Y
# IðS rk ðhÞ 4arðkÞ ; T rk ðhÞ 4brðkÞ Þ
h¼kþ1
X
ð21Þ
X
rk1 1park1 ð1Þ ooark1 ðk1Þ pcðnkþ1Þ 1pbrk1 ð1Þ oobrk1 ðk1Þ prðnkþ1Þ
E
" n Y
f rk1 ðjÞ ðX cðnjÞ;rðnjÞ Þ
IðSrðiÞ ¼ arðiÞ ; T rðiÞ ¼ brðiÞ Þ
i¼1
. ¼ ..
"
¼E " ¼E " ¼E
n Y
j¼1 n Y
n Y
# IðSrk1 ðhÞ 4arðk1Þ ; T rk1 ðhÞ 4brðk1Þ Þ
ð22Þ
h¼k
f r0 ðjÞ ðX cðnjÞ;rðnjÞ Þ
j¼1 n Y
f rðiÞ ðX arðiÞ brðiÞ Þ
i¼1
j¼k k1 Y
k1 Y
#
n Y
# IðSr0 ðhÞ 4arð0Þ ; T r0 ðhÞ 4brð0Þ Þ
h¼1
f j ðX cðnjÞ;rðnjÞ Þ # f j ðX jj Þ ;
by exchangeability.
j¼1
P Note that r denotes summation over all possible n! permutations of ð1; . . . ; nÞ: The predictability of ðSrðnÞ ; T rðnÞ Þ and the total order of ðarðiÞ ; brðiÞ Þ are used to obtain (14). Total ordering is again used to enable us to sum over arðnÞ ; brðnÞ to obtain (16). Eq. (18) follows by using the weak F-SE property after conditioning on Fc2;r2 : Since arðn1Þ1 pc 2 and brðn1Þ1 pr 2; the indicator functions in (18) are all Fc2;r2 measurable. Since IðSrn2 ðnÞ ; Srn2 ðn1Þ 4arðn2Þ ; T rn2 ðnÞ ; T rn2 ðn1Þ 4brðn2Þ Þ is Farðn2Þ 1;brðn2Þ 1 measurable Eq. (19) follows after summing over arðn1Þ ; brðn1Þ and the last two terms of r: Eq. (20) follows by successive conditioning on FarðkÞ1 ;brðkÞ1 : Finally Eq. (22) follows as in (19).
ARTICLE IN PRESS B.G. Ivanoff, N.C. Weber / Statistics & Probability Letters 70 (2004) 95–108
105
An extension to infinite arrays and unbounded predictable times follows as in the proof of Theorem 11.3 in Kallenberg (2002). & 3.3. Strong F-JE arrays The following result is the analogue of Theorem 4.1.2 in Ivanoff and Weber (2004), a predictable sampling theorem for finite and infinite strong F-JE arrays. Theorem 3.3. Let X be a finite or infinite strong n n F-JE array and suppose that t1 ; . . . ; tk are distinct, finite Fii -predictable times. Then ðX ti tj ; 1pi; jpkÞ¼D ðX ij ; 1pi; jpkÞ: Proof. We begin by assuming that no1: Since the array is square, we may proceed by using allocation sequences. Define ai ¼ inff‘; t‘ ¼ ig;
1pipn:
Initially assume that n ¼ k; so that ðt1 ; . . . ; tk Þ and ða1 ; . . . ; ak Þ are inverse random permutations of ð1; . . . ; nÞ: For each m 2 f0; . . . ; kg; put am i ¼ ai for all ipm; and define recursively m m am iþ1 ¼ minðf1; . . . ; kgnfa1 ; . . . ; ai gÞ;
mpipk 1:
m ðam 1 ; . . . ; ak Þ
m1 We have that is Fm1;m1 measurable, and whenever iom; am ¼ ai : Recall i ¼ ai Dmm ¼ fði; jÞ : iXm or jXmg: For any bounded measurable functions f hk on G, " # YY E f ami amj ðX ij Þ i
j
Y Y
¼E
iom jom
Y Y
¼E " ¼E
iom jom
YY i
j
" f ami amj ðX ij ÞE
Y
#!
Y
f ami amj ðX ij Þ j Fm1;m1
ði;jÞ2Dmm
f am1 ðX ij ÞE am1 i j
" Y
Y
#! f am1 ðX ij Þ j Fm1;m1 am1 i j
ð23Þ
ði;jÞ2Dmm
#
f am1 ajm1 ðX ij Þ : i
The conditional expectation in (23) follows from strong F-JE, noting that fam1 ; mpipkg ¼ i m k 0 fai ; mpipkg: Now, ai ¼ ai ; ai ¼ i; and as in Kallenberg (2002), we may conclude that " # " # " # YY YY YY E f ij ðX ti tj Þ ¼ E f ai aj ðX ij Þ ¼ E f ij ðX ij Þ : i
j
i
j
i
j
A monotone class argument completes the proof when k ¼ n: The extension to general values of k and infinite values of n is done as in Kallenberg (2002). &
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3.4. Weak F-JE arrays For completeness we include the following predictable sampling result for weak F-JE arrays which is an immediate consequence of the diagonal terms being conditionally exchangeable given fFii g: Lemma 3.4. Let X be a finite or infinite weak F-JE array and suppose that t1 ; . . . ; tk are distinct, finite Fii -predictable times. Then ðX t1 t1 ; . . . ; X tk tk Þ¼D ðX 11 ; . . . ; X kk Þ:
4. Infinite and Ergodic arrays We now consider situations in which exchangeability is equivalent to the weaker concept of spreadability. We recall that an array is separately (respectively, jointly) spreadable ((SS, respectively JS) if (3) (respectively, (5)) is satisfied for increasing subsequences of indices. The weak and strong F-SS and JS properties are analogous to the corresponding properties for exchangeable arrays (for details, see Ivanoff and Weber (2004)). 4.1. Infinite arrays For infinite sequences, it is well known that conditional spreadability and exchangeability are equivalent. Therefore, it is an immediate consequence of Lemma 2.1 that if X is an infinite array, X is strong F-separately spreadable (F-SS) if and only if X is strong F-SE. We continue with an infinite weak F-SS array X. We say that X is F-stationary if yST X ¼D X for every bounded adapted random time ðS; TÞ: By Theorem 1 of Ivanoff and Weber (1996), X is weak F-SE if and only if X is F-stationary so in this case, by Lemma 3.2.1 of Ivanoff and Weber (2004) the weak F-SE property is equivalent to the weak F-SS property. In the case of the joint properties, even in the case of infinite arrays joint spreadability is not strictly equivalent to joint exchangeability. If X is JE, then it is straightforward by simply permuting i and j that X ij ¼D X ji : Thus, all the diagonal elements are identically distributed as are all the off-diagonal elements, although the two distributions may be different. In the case of a JS array, the diagonal elements have the same distribution, as do the random variables ðX ij ; iojÞ and the random variables ðX ij ; i4jÞ; but the three distributions may be different. However, Kallenberg (1992) notes that in the infinite case an array is JS if and only if the array X ‘‘folded’’ on the diagonal (i.e. the upper triangular array Y :¼ðY ij ¼ ðX ij ; X ji Þ; ipjÞ) can be embedded in a JE array. 4.2. Finite Ergodic arrays An ergodic array is generated by random permutations of the rows and columns of a finite array of constants. In this situation we shall see that under certain conditions, spreadability and
ARTICLE IN PRESS B.G. Ivanoff, N.C. Weber / Statistics & Probability Letters 70 (2004) 95–108
107
exchangeability are equivalent, as are the weak and strong F properties. First we consider the SE and SS properties. Lemma 4.1. Let X be an array generated by independent random permutations p and s of the columns and rows of a (finite) c r array of distinct constants and let Fij ¼ sfpð1Þ; . . . ; pðiÞ; sð1Þ; . . . ; sðjÞg: Then the following are equivalent: 1. 2. 3. 4.
X ST ¼D X 11 for every F-predictable random time ðS; TÞ: X is weak F-SS. X is weak F-SE. X is strong F-SE.
Proof. We have that 4 ) 3 ) 2 and it was shown in Theorem 3.2.3 in Ivanoff and Weber (2004) that 2 ) 1: It remains to prove 1 ) 4: It was proven in Theorem 4 of Ivanoff and Weber (1996) that 133: If X ST ¼D X 11 for every F-predictable random time ðS; TÞ; then examining their proof, it is seen that given Fij ; the remaining column and row permutations are independently and uniformly distributed over the ðc iÞ! and ðr jÞ! possibilities, respectively. This implies the appropriate conditional distribution for the entire shell outside ðX hk ; 1phpi; 1pkpjÞ; and so it follows that X is strong F-SE. & We move now to the joint exchangeability and spreadability properties. Lemma 4.2. Let X be a finite n n array generated by applying the same random permutation to the rows and columns of an array of constants where the diagonal of the array consists of n distinct elements and the diagonal and off-diagonal entries form disjoint sets. The following are equivalent: 1. 2. 3. 4.
X SS ¼D X 11 for every Fii -predictable random time S. X is weak F-JS. X is weak F-JE. X is strong F-JE.
Proof. The key point in the JE case is that the random permutation, p; is determined by the behaviour of the diagonal elements, and so the proof is essentially one dimensional. The result then follows from Lemma 3.4 and by arguing as in the proof of Lemma 4.1. &
References Aldous, D.J., 1985. Exchangeability and related topics. In: E´cole d’e´te´ de probabilitie´s de Saint-Flour XIII. Lecture Notes in Mathematics, P.L. Hennequin (Ed.), vol. 1117. Springer, Berlin. De Finetti, B., 1929. Fuzione caratteristica di un fenomeno aleatorio. Atti Congr. Int. Mat. Bologna 1928 6, 179–190. De Finetti, B., 1930. Fuzione caratteristica di un fenomeno aleatorio. Mem. Roy. Acc. Lincei 4, 86–133. Ivanoff, B.G., Weber, N.C., 1996. Some characterizations of partial exchangeability. J. Austral. Math. Soc. (A) 61, 345–359. Ivanoff, B.G., Weber, N.C., 2004. Spreadable arrays and martingale structures. J. Austral. Math. Soc., to appear. Kallenberg, O., 1982. Characterizations and embedding properties in exchangeability. Z. Wahrsch. Verw. Gebiete 60, 249–281.
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Kallenberg, O., 1988. Spreading and predictable sampling in exchangeable sequences and processes. Ann. Probab. 16, 508–534. Kallenberg, O., 1992. Symmetries on random arrays and set-indexed processes. J. Theoret. Probab. 5, 727–765. Kallenberg, O., 2000. Spreading-invariant sequences and processes on bounded index sets. Probab. Theory Related Fields 118, 211–250. Kallenberg, O., 2002. Foundations of Modern Probability, second ed. Springer, New York. Kingman, J.F.C., 1978. Uses of exchangeability. Ann. Probab. 6, 183–197. Ryll-Nardzewski, C., 1957. On stationary sequences of random variables and the de Finetti’s equivalence. Colloq. Math. 4, 149–156.