ON COMPLETE CONVERGENCE IN A BANACH SPACE

Sufficient conditions are given under which a sequence of independent random elements taking values in a Banach space satisfy the Hsu and Robbins law of large numbers. The complete convergence of random indexed sums of random elements is also considered.


INTRODUCTION
Let {X,,, n _> } be a sequence of independent random elements taking values in a separable Banach space (B, {{).Put S, Z X,. a sequence {X,, n > 1} of random elements is said to satisfy the law of large numbers of Hsu-Robbins type if for any given > 0 Hsu nd Robbins [1] proved that the existence of the second moment of independent, identica,lly distributed rndom vribles for which EX 0, implies the Hsu-Robbins type lw of lrge numbers, grd6s [2] showed that the existence of the second moment of independent, identicMly distributed rndom vribles nd the condition EX 0 is lso the necessary one for the Hsu-Robbins type law of large numbers.Considerations concerning (1.1) for sequences nd subsequences of independent, identically distributed rndom vribles cn be found in Ktz [a], Bum, Ktz [4], Asmussen, Eurtz [] nd Out [fi].The results in those cses re given under the assumption when there exists finite moment of order r (1 < r 5 2).
Some conditions, which guarantee the convergence of (1.1) for sequences nd subsequences in the cse nonidenticMly distributed rndom variables cn be found in Duncan, SzynM [7], Br- toszyfiski Puri [8] and Kuczmaszevska,, Szynal [9], [10].For instance, it has been shown in Duncan,   Szynal [7] that if a, sequence {X.. n 2 1} of independent random variables with EX 0 and (,) () (,') . - The following eample shows that the assumptions (i)-(iv) hich are sucient conditions ir (I.I) in the cse of independent random variables re not sucient if e conser sequences of independent random elements taking values in Banach space B. EXAMPLE.Let denote the separable Banach space and e" denote the elenent having for its n-th coordinate and 0 in the other coordinates.
LEMMA 1.Let {X,, n > 1} be a sequence of independent random elements taking values in a real separable Bana,ch space (B, II) with a symmetric distribution.Then for every j=l, 2,...,n and > 0 (2.1)Moreover, we shall use the following lemmas.
whew .V is a svmel rized version of X.
COROLLARY 1.Let {X,, n > be a sequence of independent, symmetrically distributed, B-valued random elements.Suppose that {nk, k > 1} is a strictly increasing sequence of positive integers.If for some positive integer and any given e > 0 .Now we consider the Hsu and Robbins law of large numbers for subsequences of independent, nonsymmetrically distributed random elements taking vahles in a real separable Banach space.THEOREM 2. Let {X,, n > 1} be a sequence of independent, B-valued random elements.Suppose that {n, k _> is a strictly increasing sequence of positive integers.If for some positive integer and any given . .
Nov it is easy to show that there exists some positive integer j, for which k=l k=l c. ,'-")'(EIIXII)' < .
To remove the symmetry assumption we proceed similar as it has been done in the proof of Theorem 2.
The following corollary is an extension of Adler's result to independent non-identically distributed B-valued random elements.COROLLARY 7. Let {X,, n >_ be a sequence of independent, B-valued random elements and {T., n > be positive integer valued random variables.Suppose that {a., n > is a strictly increasing sequence of positive integers and {/3,,, n > 1} is a sequence of positive constants such that a.The next corollary is an extension of one of the results given in Adler [15] to the case of i.i.d.B-valued random elements.