Limit Theorems for Exchangeable Random Variables via Martingales

Transcription

1 Limit Theorems for Exchangeable Random Variables via Martingales Neville Weber, University of Sydney. May 15, 2006 Probabilistic Symmetries and Their Applications

2 A sequence of random variables {X 1, X 2,.., X N } is said to be exchangeable if {X i } D = {X π(i) } for every permutation π of {1, 2,.., N}. An infinite sequence is said to be exchangeable if every finite subset is exchangeable. For an infinite exchangeable sequence if we condition on an appropriate σ-field then the sequence behaves like a sequence of independent and identically distributed random variables (de Finetti (1930)). Blum et al. (1958), Bühlmann (1958) used this result to obtain the central limit theorem for sums of exchangeable random variables. For an array of random variables where each row consists of finitely exchangeable variables the above technique does not apply. Martingale methods provide a unified approach to both situations. The martingale (or reversed martingale) idea was suggested by Loynes (1969) in the context of U-statistics and developed by Eagleson (1979, 1982), Eagleson and Weber (1978), and Weber (1980) for exchangeable variables. Probabilistic Symmetries and Their Applications 1

3 Martingales Let F = {F n } be an increasing sequence of σ-fields, i.e. F n F n+1 and let S 1, S 2,.. be a sequence of random variables such that S i is F i -measurable. F n can be thought of as the history of the process up to time n. {S n } is a martingale with respect to F if E S n <, and E(S n F m ) = S m for all m < n. Martingales include sums of zero mean, i.i.d. random variables, S n = Σ n i=1 X i, and many of the properties of sums carry across to martingales under suitable conditions. Flexibility in the choice of F n makes martingale methods very useful. We will show that we can exploit the symmetry of various exchangeable situations to construct σ-fields that allow us to form martingales and to evaluate key conditional expectations. Probabilistic Symmetries and Their Applications 2

4 Let {Y nj } be a triangular array of martingale differences with respect to {F nj }. Theorem 1. [Scott (1973), Theorem 2] Suppose (i) P n EY 2 nj 1, (ii) max j n Y nj (iii) P n Y 2 nj p 0, and p 1. Then S n = D Y nj N(0, 1). Probabilistic Symmetries and Their Applications 3

5 A Poisson convergence result follows from the results for infinitely divisible laws developed by Brown and Eagleson (1971). See also Freedman (1974). Let {A nj } be an array of events with A nj being F nj -measurable. Theorem 2. If, for some constant λ, (i) P n P (A nj F n,j 1 ) p λ, and (ii) max j n P (A nj F n,j 1 ) p 0, then T n = I(A nj ) D Poisson(λ). Probabilistic Symmetries and Their Applications 4

6 Exchangeable Random Variables Consider an array of row-wise finitely or infinitely exchangeable random variables {X nj : j = 1,.., m n ; n = 1, 2,..}. Construct an array of σ-fields such that X nj is F nj -measurable and F n,j 1 F nj, j = 1, 2,.. Form a martingale difference array and apply Theorem 1 to obtain results for S n = [X nj E(X nj F n,j 1 )]. For exchangeable sequences we can choose F nj so that we can evaluate E(X nj F n,j 1 ). Thus we can obtain conditions under which P n X nj converges in distribution to normal. Specifically, let F n,j 1 = σ{x n1,.., X n,j 1, Xmn i=j X ni } = σ{x n1,.., X n,j 1, Xmn i=1 X ni } Probabilistic Symmetries and Their Applications 5

7 Xmn i=j Xmn X ni = E( X ni F n,j 1 ) = = Xmn i=j Xmn i=j i=j E(X ni F n,j 1 ) E(X nj F n,j 1 ), by exchangeability = (m n j + 1)E(X nj F n,j 1 ). Thus Xmn E(X nj F n,j 1 ) = (m n j + 1) 1 i=j X ni. Probabilistic Symmetries and Their Applications 6

8 Theorem 3. Given {X ni } as above with (i) nex 2 n1 1, (iv) P n X2 nj p 1, (ii) n 2 EX n1 X n2 0, (v) n/m n 0. (iii) p max j n X nj 0, Then S n = D X nj N(0, 1). Proof. Let Y nj = X nj E(X nj F n,j 1 ). EY 2 nj = EX 2 nj E( E 2 (X nj F n,j 1 )). Probabilistic Symmetries and Their Applications 7

9 E( E 2 (X nj F n,j 1 )) = EX 2 n1 /(m n j + 1) + m n j m n j + 1 EX n1x n2 EX 2 n1 log(n + 1) + n EX n1x n2 0, by (i) and (ii). and max j n Y nj max j n X nj + max j n E(X nj F n,j 1 ) max j n E(X nj F n,j 1 ) [ p so (i), (ii) and (iii) imply max j n Y nj 0. Similarly we have P n Y 2 p nj 1. Thus E 2 (X nj F n,j 1 )] 1/2 p 0, D Y nj N(0, 1). Probabilistic Symmetries and Their Applications 8

10 To obtain the final result we need to show that E(X nj F n,j 1 ) p 0. By exchangeability E(X nj F n,j 1 ) = E(X nm n F n,j 1 ) and so for fixed n, {E(X nm n F n,j 1 ), F n,j 1 } is a martingale. Given ɛ > 0 P ( P n E(X nj F n,j 1 ) > ɛ) P (max k n P k E(X nj F n,j 1 ) > ɛ) P (n max k n E(X nm n F nk ) > ɛ) (n/ɛ) 2 E[E(X nm n F nn )] 2, by Kolmogorov s inequality (n/ɛ) 2 [E(X 2 n1 )/(m n n + 1) + EX n1 X n2 ] 0 by (i) and (ii). Probabilistic Symmetries and Their Applications 9

11 Let G nj be the σ-field of measurable sets involving {X n1,.., X nm n} that are unaffected by permuting of the indices 1, 2,.., j, i.e. G nj = σ{(x (1) n1,.., X(j) nj ), X n,j+1,.., X nm n} where (X (1) n1,.., X(j) nj ) denotes the order statistics of the first j terms in the nth row. Here G nj G n,j+1, and, by symmetry, E(X n1 G nj ) = j 1 jx X ni = j 1 S nj. i=1 Thus {j 1 S nj, G nj } is a reversed martingale for each n. Probabilistic Symmetries and Their Applications 10

12 Let Z nj = E(X n1 G nj ) E(X n1 G n,j+1 ) = j 1 {E(X n1 G n,j+1 ) X n,j+1 } so P m n 1 j=n Z nj = E(X n1 G nn ) E(X n1 G nm n) = n 1 P n X nj E(X n1 G nm n). and If we have an exchangeable array with non-random row sums, P mn X nj = 0, say, n 1 X nj = mn 1 X j=n Z nj. Probabilistic Symmetries and Their Applications 11

13 Thus we can represent P n X nj as a sum of reversed martingale differences. The martingale CLT can be applied to the array {Z n,m n j+1, G n,m n j+1} m n n+1 to get the following variation of Chernoff and Teicher (1958). Theorem 4. Let {X ni } be an array of row wise exchangeable random variables with P m n i=1 X ni = 0 for each n. If (i) n/m n α as n, α [0, 1) p (ii) max i m n X ni 0, (iii) P n i=1 X2 p ni 1, as n, then S n D N(0, 1 α). Probabilistic Symmetries and Their Applications 12

14 Similar methods can be applied to arrays of row wise exchangeable events to obtain Poisson limit laws. Let H nj = σ{a n1,.., A nj, P mn i=j+1 I(A ni)} then P (A nj H n,j 1 ) = (m n j + 1) 1 P mn i=j I(A ni). Applying Theorem 2 we obtain Theorem 5. [Ridler-Rowe (1967)] If (i) np (A n1 ) λ, (ii) n 2 P (A n1 A n2 ) λ 2 (iii) n/m n 0 then I(A nj ) D Poisson(λ). Probabilistic Symmetries and Their Applications 13

15 To establish the result note E P (A nj H n,j 1 ) P (A nj ) 2 = E(I(A nj ) Xmn i=j I(A ni )/(m n j + 1) P (A nj ) 2 ) = P (A n1 )/(m n j + 1) + m n j m n j + 1 P (A n1 A n2 ) P (A n1 ) 2 so E P (A nj H n,j 1 ) P (A nj ) n (m n n) 1/2P (A n1) 1/2 + n P (A n1 A n2 ) P (A n1 ) 2 1/2 0, using (i) - (iii). Probabilistic Symmetries and Their Applications 14

16 Contractable Sequences Kallenberg (2000) renewed interest in a more general structure than exchangeability. A sequence of random variables (X 1,..., X n ) taking values in some measurable space (G, G) is contractable (sometimes referred to as spreadable) if for any finite m < n and increasing subsequence of indices k 1 <... < k m, (X k1,..., X k m) = D (X 1,..., X m ). If the above is true for any finite subset of distinct, but not necessarily increasing indices, then the sequence is exchangeable. Contractability implies the variables are identically distributed. Exchangeability and contractability are equivalent in the infinite case (Ryll-Nardzewski (1957)) but not in the finite case. What extra features or conditions are needed for contractable sequences to be exchangeable? Probabilistic Symmetries and Their Applications 15

17 Example (Kallenberg (2000)) (X 1, X 2, X 3 ) takes values in {0, 1, 2}. Joint distribution: Value Probability Value Probability (0,0,1) 2/14 (1,2,2) 1/14 (0,1,2) 2/14 (2,0,0) 2/14 (0,2,0) 2/14 (2,0,2) 1/14 (1,0,2) 1/14 (2,1,0) 1/14 (1,2,0) 1/14 (2,2,1) 1/14 P (X i = 0) = 6/14, P (X i = 1) = 3/14, P (X i = 2) = 5/14. Note (X 1, X 2 ) D = (X 1, X 3 ) D = (X 2, X 3 ), but (X 1, X 2 ) D (X 2, X 1 ) (eg P (X 1 = 2, X 2 = 0) = 3/14 whereas P (X 1 = 0, X 2 = 2) = 2/14). Probabilistic Symmetries and Their Applications 16

18 F - Contractability Given a discrete filtration F = (F 0, F 1,...), the finite or infinite sequence (X 1, X 2,..., ) is F-contractable (exchangeable) if it is adapted to F and when conditioned on F k, θ k X = (X k+1,...) is contractable (exchangeable) for every k. Every contractable sequence is F-contractable wrt the minimal filtration. If X is F-contractable then E(X n F k ) = E(X k+1 F k ), for all n > k. Probabilistic Symmetries and Their Applications 17

19 Specifically if X 1,.., X n are F-contractable where F j 1 = σ{x 1,.., X j 1, X i } = σ{x 1,.., X j 1, i=j X i } i=1 then E(X j F j 1 ) = (n j + 1) 1 X i. This was a key observation in applying martingale weak limit results to obtain results for exchangeable sequences. Lemma 1. [Theorem 1.13, Kallenberg (2005)] If X = (X 1, X 2,.., X n ) is F-contractable and Σ n i=1 X i is F 0 -measurable then X is exchangeable given F 0. i=j Probabilistic Symmetries and Their Applications 18

20 Partially Exchangeable Arrays. U-statistics Given a sequence of iid random variables ξ 1, ξ 2,.. and a function h : R m R which is symmetric in its arguments, the U-statistic with kernel h (of degree m) is U n = n 1X h(ξi1,.., ξ i m), m n where P n denotes summation over 1 i 1 < i 2 <.. < i m n. Examples: 1. Sample moments h(x) = x k 2. Sample variance h(x, y) = 1 2 (x y)2 3. Wilcoxon statistic h(x, y) = I(x + y > 0) 4. Rayleigh statistic h(x, y) = 2 cos(x y), x [0, 2π) Probabilistic Symmetries and Their Applications 19

21 Let I n = σ{(ξ (1),.., ξ (n) ), ξ n+1, ξ n+2,..}. Then I n I n+1 and U n = E(h(ξ 1,.., ξ m ) I n ). Thus {U n } is a reversed martingale wrt {I n }. If E h(ξ 1,.., ξ m ) < strong consistency follows from the reversed mg convergence theorem. Loynes (1969) conjectured that the CLT for U-statistics, established by Hoeffding (1948), should follow via martingale techniques. Probabilistic Symmetries and Their Applications 20

22 Jointly Exchangeable Arrays U-statistics are special cases of partial sums of jointly exchangeable random variables. The array {X i1,i 2,..,im} is jointly (or weakly) exchangeable if {X i1,i 2,..,im} D = {X π(i1 ),π(i 2 ),..,π(im)} (1) where π is a permutation of the positive integers that leaves all but a finite number of terms fixed. Trivially X i1,i 2,..,im = h(ξ i1, ξ i2,.., ξ i m) satisfies (3). The arrays {h(ξ i1,.., ξ i m)} are infact dissociated arrays in that terms with disjoint indexing sets are independent. Limit theorems for dissociated arrays were established by Silverman (1976). Probabilistic Symmetries and Their Applications 21

23 For the infinite, symmetric, jointly exchangeable array case Eagleson and Weber (1978) used n 1X I n = σ{t n, T n+1,..} where T n = E(X 1,2,..,m I n ) = Xi1,i m 2,..,im. {T n, I n } is a reversed martingale and martingale methods were used to establish the central limit theorem for T n. n k (k, k) (0,0) k Figure 1: σ-field structure for m = 2 Probabilistic Symmetries and Their Applications 22

24 Aldous (1981) extended de Finetti s result to arrays and proved that every infinite, jointly exchangeable array is a mixture of dissociated arrays. Thus the central limit result for partial sums can be deduced from this and Silverman (1976). Aldous representation does not hold for finitely jointly exchangeable arrays. Martingale methods are particularly powerful in these cases. Results obtained for sequences of partial sums of finitely, jointly exchangeable random variables have direct application to U-statistics based on samples drawn without replacement from finite populations. To establish limit theorems we need to select filtrations that enable key conditional expectations to be evaluated. Probabilistic Symmetries and Their Applications 23

25 Eagleson (1979) constructed an appropriate filtration to create forward martingales to establish Poisson limit laws. Theorem 6. [Eagleson (1979), Theorem 3] Let {A (n) ij : 1 i, j m n } be a sequence of symmetric, jointly exchangeable events. If, for some constant λ, (i) `n (n) 2 P (A 12 ) λ, (ii) n 3 P (A (n) 12 A(n) 13 ) 0, (iii) `n 2P (n) 2 (A 12 A(n) 34 ) λ2, (iv) n/m n 0, then X 1 i<j n I(A (n) ij ) D Poisson(λ). Probabilistic Symmetries and Their Applications 24

26 k A (n) ij (k, k) (0,0) k Figure 2: σ-field structure The σ-field structure used in this case was F nk = σ{a (n) ij m n, 1 i, j k; U (n) 1k,.., U (n) kk } where U (n) αk = P mn β=k+1 I(A(n) αβ ), α k. F n,k 1 F nk. For i < j P (A (n) ij F n,j 1) = U (n) i,j 1 /(m n j + 1). Probabilistic Symmetries and Their Applications 25

27 Symmetric statistics of U-type By considering sequences of arrays we can obtain results for generalizations of U- statistics where the kernels depend on the sample size. Such statistics arise naturally in analysing spatial data. Given a sequence of iid random variables ξ 1, ξ 2,.. define S n = n 1X hn (ξ i1,.., ξ i m). m n Asymptotic normality can be established via reversed martingale methods using the σ-fields F nj = σ{(ξ (1),.., ξ (j) ), ξ j+1, ξ j+2,..} (Weber (1983)). Poisson limits were obtained by Silverman and Brown (1978). Probabilistic Symmetries and Their Applications 26

28 Examples Interpoint distance statistic - test for clusters h n (ξ 1, ξ 2 ) = I(d(ξ 1, ξ 2 ) βn α ) for some constants α and β, ξ i R k. A normal limit exists for if α [0, 2/k) and a Poisson limit if α = 2/k. Test for collinearity. Kendall and Kendall (1980) suggest using an indicator function h n and counting the number of triples where the largest interior angle is greater than π βn α. If 0 α < 2 then a normal limit exists. Probabilistic Symmetries and Their Applications 27

29 Separately Exchangeable Arrays {X (n) ij : 1 i c n, 1 j r n } D = {X (n) πn(i),σn(j) } where σ n is any permutation of (1, 2,.., c n ) and π n is any permutation of (1, 2,.., r n ). In these cases the arrays are rectangular and so typically when constructing the σ-fields we have to consider row sums, column sums plus the total of the upper rectangular region. Exchangeability within rows and within columns is exploited to evaluate the conditional expectations. Probabilistic Symmetries and Their Applications 28

30 r n j (i, j) (0,0) i c n Figure 3: σ-field structure for separately exchangeable arrays. Probabilistic Symmetries and Their Applications 29

31 This approach can also be used to construct relevant 1- and 2- martingales to obtain 2-d limits. For example, Theorem 7. [Brown et al. (1986), Theorem 4.1] Let {A (n) ij : 1 i r n, 1 j m n } be a sequence of separately exchangeable events. If, for some constant λ, (i) n 2 P (A (n) 11 ) λ, (ii) n 3 P (A (n) 11 A(n) 12 ) 0, (iii) n 3 P (A (n) 11 A(n) 21 ) 0, (iv) n 4 P (A (n) 11 A(n) 22 ) λ2, (v) n/m n 0, and n/r n 0 then [ns] X i=1 [nt] X I(A (n) ij ) converges in distribution to a Poisson process with intensity λ times Lebesgue measure. Probabilistic Symmetries and Their Applications 30

32 References [1] Aldous, D.J. (1981) Representations for partially exchangeable arrays of random variables. J. Multivariate Anal., 11, [2] Blum, J.R., Chernoff, H., Rosenblatt, M. and Teicher, H. (1958). Central limit theorems for interchangeable processes, Canad. J. Math., 10, [3] Brown, B.M. and Eagleson, G.K. (1971) Martingale convergence to infinitely divisible laws with finite variances. Trans. Amer. Math. Soc., 162, [4] Brown, T.C., Ivanoff, B.G. and Weber, N.C. (1986) Poisson convergence in two dimensions with application to row and column exchangeable arrays. Stoch. Proc. Appl., 23, [5] Bühlmann, H. (1958) Le problème limit central pour les variables alèatoires échongeables. C.R. Acad. Sci. Paris, 246, [6] Chernoff, H. and Teicher, H. (1958) A central limit theorem for sums of interchangeable random variables. Ann. Math. Statist., 29, [7] De Finetti, B. (1930) Funzione caratteristica di un fenomeno aleatorio. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 4, [8] Eagleson, G.K. (1979) A Poisson limit theorem for weakly exchangeable events. J. Appl. Probab., 16, [9] Eagleson, G.K. (1982) Weak limit theorems for exchangeable random variables. In Exchangeability in Probability and Statistics Ed. G. Koch and F.Spizzichino, North-Holland, Amsterdam-New York, [10] Eagleson, G.K. and Weber, N.C. (1978). Limit theorems for weakly exchangeable arrays. Math. Proc. Camb. Philos. Soc., 84, [11] Freedman, D. (1974) The Poisson approximation for dependent events. Ann. Probab., 2, [12] Hall, P. and Heyde, C.C (1980) Martingale Limit Theory and Its Application. Academic Press, London [13] Hoeffding, W. (1948) A class of statistics with asymptotically normal distribution. Ann. Math. Statist., 19, Probabilistic Symmetries and Their Applications 31

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