STRONG LAWS OF LARGE NUMBERS FOR ARRAYS OF ROWWISE CONDITIONALLY INDEPENDENT RANDOM ELEMENTS 1

Let Xnk} be an array of rowwse conditionally independent random elements in a separable Banach space of type p, 1 _< p _< 2. Complete convergence of n-I/r Xnk to 0, 0 < r < p _< 2 is obtained k=l by using various conditions on the moments and conditional means. A Chung type strong law of large numbers is also obtained under suitable moment conditions on the conditional means.


I. INTRODUCTION AND PRELIMINAPdES
Let (, I1" II)   real separable Banach space.Let (f,at, P) denote a probability space.A random element (r.e.) X in g is a function from ft into g which is A-measurable with respect to the Borel subsets B($).The rth absolute moment of a random element X is E II X 11 r where E is the expected value of She random variable I I X I I r.The expected value of 1Received: August, 1992.Revised: January, 1993.RONALD F. PA'ITERSON, ABOLGHASSEM BOZORGNIA and ROBERT L. TAYLOR a random element X is defined to be the Bochner integral (when E I I X I I < oo) and is denoted by EX.The concepts of independence and identical distributions for real-valued random variables extend directly to g.A separable Banach space is said to be of (Rademacher) type p, 1 _< p _< 2, if there exist a constant C such that E I I Xu I! -< C E il X I I (1.1) k=l k=l for all independeng random elements X,..., X n with zero means and finite pth moments.The sequence of random elements {Xn) is said to be conditionally independent if there exists a sub- r-field ( of A such that for each positive integer m where IX BI] denotes the conditional probability of the random element X being in the Borel set B given the sub-o'-field '. independent random elements are conditionally independent with respect to the trivial r-field Throughout this paper {Xnk: 1 < k <_ n, n > 1} will denote rowwise conditionally independent random elements in such that EXnk = 0 for all n and k.The first major result of this paper shows that 1 XnkO completely dl/n k=l where complete convergence is defined (as in Hsu and Robbins [5]) by for each e > 0.
which provides (1.2) The second major result is a Chung type strong law of large numbers (SLLN) k=l where a n < a n +1 and ldmooa n = oo.For comparisons with (1.2) and (1.4), a brief partial review of previous results will follow.
Erdgs [4] showed that for an array of i.i.d, random variables {Xnk}, (1.2) holds if and only if E IX11 2r < oo.Jain [8] obtained a uniform SLLN for sequences of i.i.d.r.e.'s in a separable Banach space of type 2 which would yield (1.2) with r = 1 for an array of r.e.'s {Xnk}.Woyczynski [12] showed that 1 X/,--.0completely (1.5) ,i;-< = for any sequence {Xn} of independent r.e.'s in a Banach space of type p, 1 _< r < p _< 2 with EXn=O for all n which is uniformly bounded by a random variable X satisfying E IX[r< oo.Recall that an array {Xnk) of r.e.'s is said to be uniformly bounded by a random variable X if for all n and k and for every real number > 0 I I X.kI I > t] < P[iXl > t]. (1.6) Hu, Moricz and Taylor [7] showed that ErdSs' result could be obtained by replacing the i.i.d.condition by the uniformly bounded condition (1.6).Taylor and Hu [9] obtained complete convergence in type p spaces, 1 < p < 2 for uniformly bounded, rowwise independent r.e.'s.
Bozorgnia, Patterson and Taylor [1] obtained a more general result by replacing the assumption of uniformly bounded random elements with moment conditions.One complete convergence result of this paper, given in Section 2, is obtained by assuming a condition on the conditional means and extends the result in Bozorgnia et.al [1].
If {Xn} is a sequence of independent (but not necessarily identically distributed) r.v.'s, Chung's SLLN yield (1.4) for r.v.'s if @(t) is a positive, even, continuous function such that either (t) as tlrc (1.7) and E@(Xn) (a': where {an} is a sequence of real numbers such that a n < (1.9) = 0 and (1.8) holds where 1" and denote monotone increasing and monotone decreasing respectively.
Wu, Taylor and Hu [6] considered SLLN's for arrays of rowwise independent random variables, {Xni: 1 < < n, n > 1}.They obtained Chung type SLLN's under the more general conditions: (Itl), (ltl tl and where @(t) is a positive, even function and r is a nonnegative integer, where k is a positive integer and {an} is a sequence of positive real numbers defined in (1.4).
for some nonnegative integer r, where the separable Banach space is of type p, 1 < p < 2, (1.15) where k is a positive integer.

EXni = O,
(1.16) Z E(( I I I I )) (1.17) In Section 2 of this paper, SLLN's for arrays of rowwise conditionally independent r.e.'s are obtained for Banach spaces under conditions similar to those of Chung [3], Hu, Taylor  and Wu [6] and Bozorgnia, Patterson and Taylor [2] with appropriate conditions on the conditional means.These new results address the question of possible exchangeability extensions in the affirmative, and in addition, provide a class of new results for conditionally independent random elements.A generic constant, C, will be used throughout the paper.

MAIN RESULTS
A lemma by Wozczynski [12], is crucially used in the proofs of the major results, Theorems 2.2 and 2.3, and is stated here for future reference.
Lemma 2.1: Let 1 <_ p < 2 and q > 1.The following properties are equivalent: (i) The separable Banach space, , is of type p.
(ii) There exist a constant C such that for all independent r.e.'s X:,...,X n in with EX = O, and E II X II q < oo,1,2,...,n E II x II q <_ CE I I x I I p //! "= i=1 The constant C depends only on the Banach space $ and not on n.Moreover, throughout this section C will denote a generic constant which is not necessarily the same each time used but is always independent of n.
Theorem 2.2: Let {Xnk} be an array of rowwise conditionally independent random elements in a separable Banach space of type p, 1 < p < 2. If (,) ..,, II II = >_O +,,O <," l<k<n and (ii) for all 0 > O, n=l k=l where E is the conditional expectation with respect to an appropriate a-field that gives conditional independence, then 1 XnkO completely.nl/r k=l Proof: Let e > 0 be given.By Markov's inequality, P I I n-iT;/ x. II > <-e I I na/, (x.-EX.)II> = C _' n -u(r-) +c' < The second term of (2.1) is finite by (b/.aThus, the result follows./// Remark 1: Condition (ii) can be replaced by the condition E 11EcXnl I[ o = O(n-#), / > 2r-p if the r.e.'s r" are conditionally i.i.d, or rowwise infinitely exchangeable.If 0 < r < 1, and p/r > 2, then fl can be nonpositive and the bound for each row can increase.lmark 2: If the random elements are independent with zero means, then condition (ii) is identically zero when ff is chosen to be the trivial tr-field, {,A}.Thus, Theorem 2.2 generalizes the results of Bozorgnia et.al [1].
Remark 3: Condition (i) implies condition (6.2.2) in Theorem 6.2.3 of Taylor, Dafter and Patterson [10].Condition (6.2.2) was given as: E( I I Xnl I I Oq) Letting u = pq and r = 1, it follows that the third inequality in the proof of Theorem 2.2 is majorized by nt,/r_ C n -q(p-1) +c n=l which is a substantial improvement of Theorem 6.2.3 of Taylor et.al [10].Moreover, Condition (6.2.1) of Theorem 6.2.3 in Taylor et.al [10] implies Condition (ii) of Theorem 2.2 since for {Xnk} rowwise infinitely exchangeable and r = 1,.
The next result is a Chung type SLLN for arrays of rowwise conditionally independent r.e.'s in a separable Banach space of type p, 1 _< p _< 2. Let {an} be a sequence of positive real numbers such that a n < a n + l and lim a_ cx.defined in (1.115).
Let 9(t) be the positive, even function Theorem 2.3: Let {Xni) be an array of rowwise, conditionally independent random elements in a separable Banach space of type p, 1 < p < 2 such that EXni = 0 for all n and i.Let t(t) satisfy (1.15) for some r > 2. If {an) is a sequence of positive real numbers such that a n < a n + 1 and lira a n = c and if and if for some positive integer k Z E il I I p < , n--'l i=1 then Condition (1.17) implies that a-'-.where the sum ]* is over all choices of nonnegative integers Sl,...,sn such that i 1 s = k(r + 1).Now (2.9) can be shown to be summable with respect to n following the same steps as in the proof of Theorem 2.2 of Bozorngia et.al [2] for the case sip >_ r + 1 for at least one s i.The case sip < r + 1 for all is accomplished by using (2.4) instead of (1.18).Hence, the result follows.
/// Remark 4: Theorem 2.2 extends the random variable result in Hu, Taylor and Wu [6] for p 2 and the random element results in Bozorgnia, Patterson and Taylor [2] to the class of conditionally independent random variables and random elements.Again, if the r.e.'s are rowwise independent with zero means, then Condition (2.3) is equal to zero via the trivial r- field, and (2.4) becomes (1.18) with the trivial a-field.