THE LAW OF THE ITERATED LOGARITHM FOR EXCHANGEABLE RANDOM VARIABLES

In this note, necessary and sufficient conditions for laws of the iterated logarithm are developed for exchangeable random variables.

variables under certain boundedness conditions.Hartman and Winter in 1941 verified that the LIL is universally true for i.i.d, random variables when the second moment exists.There are certain extensions of the LIL to martingales.However, there appears to have been no discussions on this problem for exchangeable random variables.We address this problem in this paper and extend the LIL to exchangeable random variables with necessary and sufficient conditions for the LIL in terms of conditional mean and variance.
Random variables (r.v.'s) X1," ,X, are said to be exchangeable if the joint distribution of X1,-.-,X,, is permutation invariant.A sequence of r.v.'s {X,} is said to be exchangeable if every finite subset of the sequence is exchangeable.Obviously, i.i.d, random variables are exchangeable, but not vice versa.The LIL is said to hold for a sequence of r.v.'s {X,} with EX, 0 for all n if I n P imsup = where S EX and log denote the natural log to the base e.The following example shows that the LIL can fail even for exchangeable r.v.'s while a sequence of exchangeable r.v.'s may satisfy the LIL and not be independent r.v.'s.EXAMPLE 1.Let {X,,n >_ 1} be a sequence of i.i.d, random variables such that EX 0 and EX and let Y,, ZX,, n > 1, where the random variable Z is independent of the sequence {X,,n >_ 1} with P(Z a)= P(Z b)=0.5.It is not difficult to see that {Y,} is a sequence of exchangeable r.v.'s.Ifa=2andb=0, thenEY,,=0andEY=2foreveryn>l.We defineS2,= EY2jand U2, 2 loglog S. Clearly, P limsup =1 =0.hP lira =0 (1.1) in view of the fact that by [4], n limsup limsup /2n Io9Io9 n 2n Io91o9 n 1, a.s. (1.2) In this case, the LIL is almost nowhere true for the sequence of exchangeable random variables {Y,} versus the LIL holding for the sequence of i.i.d, random variables {X,,}.However, if a and b -1, then EY2, 1, S n, and U 2 loglog n, which yields from (1.2) P limsup Yj/S,U, ( ) 0.5 P limsup X/vt2n loglog n + 0.5 limsup y] (-X.i)lv/2n loglog. 1 1.
(1.3)This is another case where the LIL holds for exchangeable r.v.'s{Y, n > 1} which might definitely not be a sequence of independent r.v.'s as long as P(X < a)P(X, < b)+ P(X > -a)P(X, > -b) P(X, < a)P(X, > -a)+ P(Xx < b)P(X, > -b).
A similar example can be constructed to show that under certain conditions the LIL holds for martingales but fails for exchangeable r.v.'s and vice versa.Thus, conditions for the LIL to hold may be very different for exchangeable r.v.'s than for independent r.v.'s or martingales.
Necessary and sufficient conditions for the LIL to hold for exchangeable r.v.'s are established in the next section.
Below we establish the LIL and give the necessary and sufficient conditions for exchangeable r.v.'s to satisfy the LIL by using de Finetti's theorem.Let q denote the collection of distribution functions on R (real numbers) and provide with topology of weak convergence of distribution functions.Then, de Finetti's theorem [2] asserts that for an infinite sequence of exchangeable r.v.'s {X,,} there exists a probability measure # on the Borel a-field Z of subsets of such that P{g(X,, .,X,,)e B} I PF g[X,, .,X,]B} d(F) (1.4) for any B and any Borel function g:R"--R,n > 1.Moreover, PRIg(X1,...,X,,) B] is computed under the assumption that the sequence of r.v.'s {X,} is i.i.d, with common distribution function F, where EFg(X,) is the conditional mean obtained by integrating g(x) with respect to P F given by (1.4).
From (1.4), we know that if {X,} is a sequence of exchangeable r.v.'s on (F,.A,P), then {EFg(X,) is a sequence of random variables on (O,E,#) and for each f q) given, {X,,} are independent, identically distributed.
Taylor and Hu (1987) showed that for a sequence of exchangeable r.v.'s {X,} such that EF[X[ < o u a.s.EFXI 0 u-a.s.if and only if n 1--XO a.s.k=l Moreover, it was observed that EFX --0 #-a.s. is equivalent to E(X,X2)= 0. Blum, Chernoff, Rosenblatt, and Teicher (1958) showed that for a sequence of exchangeable r.v.'s {X,} such that EX < o if and only if n Xk converged in distribution to a N(O,a) r.v.v/k EFX --0 #-a.s. and EFX a #-a.s.
(1.12) 1 { for any c > 1' for each F fi q, for any c< when 0 < a F < by the LIL.The above with I/F 0 and a F a, p-a.s., confirms (1.10) and (1.11) and hence establishes (1.6).
We remark that the conditions of Theorem are satisfied of a and b 1, but are not satisfied if a 2 and b 0. EXAMPLE 2. Let X be a random variable with EX 0 and 0 < EX=< x), and let X, X,n >_ 1.Then (1.7) and (1.6) clearly fail for the exchangeable sequence {X,,n >_ 1}.
For a sequence of random variables {X,,n >_ 1}, let T be the tail a-field defined by T a(X3: J >n) and let n--1 n T limsup X / v/2n loglog n. (1.17) When {X,,n >_ 1} is a sequence of i.i.d.r.v.'s such that EX 0 and EX a=, T is almost surely equal to the constant a. Theorem also yields T a a.s.if condition (1.6) holds for exchangeable random variables, and the example in Section shows that Theorem 1 may be obtained for non-independent random variables.It is also worth observing that for exchangeable r.v.'s condition (1.7) is the necessary and sufficient condition for n-1/= X3 to converge in 3=1 distribution to a N(O,a2) r.v.It is possible for n-1/ X3 to converge in dastribution to a 3=1 mixture of normal distributed r.v.'s (cf: Chapter 2 of Taylor, Dafter, and Patterson).For example, if {X,, n >_ 1} is a sequence of exchangeable r.v.'s with EXa =0, EX < o, and E(X,X=) 0 (equivalently UF 0 #-a.s.), then n-/= X converges in distribution to a r.v.

3=1
Z which has distribution function f(x)= (a-lx)dG(a) where (I)is the standard normal 0 distribution function and G is a distribution function with support contained in [0,c).Theorem 2 provides a LIL for this setting.
THEOREM 2. If {X,, n _ 1} is a sequence of exchangeable r.v.'s with EX < cx, then in (1.17) T is an extended random variable which can be defined by REMARK.Traditionally, hypotheses of limit theorems for exchangeable random variables are phrased in terms of F EF(X) and aF EF(XI)-'F which are random variables on the probability space ((I),E,#).It can be shown that g(w)= P(Xl <_ t[T)(w)is a measurable mapping of (,A) into ((I),E) and # can be identified with the induced probability measure Pg where T is any a-field which make the exchangeable r.v.'s {X,,n 1} conditionally i.i.d. (e.g., T could be the tail a-field).Hence, T can be identified with TF, a r.v. on (I,,E,#) defined by on {F: u F > 0} Tv av on {F:uf 0}. (1.19) --c on {F:a f < 0} and the proof of Theorem 2 follows from the proof of Theorem 1.Note that T TFog a.s.
where o denotes the composition mapping.
PROOF OF THEOREM 2. Since EX < c, u F, and aF exist for / almost every F E .
From (1.12) if follows that PF limsup (X,-UF) / v/2n loglog n a F 1, L.-O 3=1 for # almost every F E @.The proof then follows by observing that completing the proof.I-I From the proof of Theorem 2, it is clear that the hypothesis EX < cx can be replaced with EFX < c #-a.s.