ON STRONG LAWS OF LARGE NUMBERS FOR ARRAYS OF ROWWISE INDEPENDENT RANDOM ELEMENTS

Let {X,} be an array of rowwise independent random elements in a sep- arable Banach space of type r, 1 _< r _< 2. Complete convergence of n )/p X. to 0 k=l

A random element X in is a function from t into which is 4 measurable with respect of the Borel subsets B().The p absolute moment of a random element X is EIIXll where E is the expected value of the random variable IIXII p.The expected value of a random element X is defined to be the Bochner integral (when EIIXII < o) d i denoted by EX.The concepts of independence and identical distributions for real-valued random variables extend directly to '.A separable Banach space is sid to be of (Rademacher for all independent random elements X1,..., X, with zero means and finite r 'h moments.Every separable Hilbert space and finite dimensional Banach space is of type 2. Every separable Banach space is at least type 1 while and L spaces are of type rain(2, r) for r>l.
Throughout this paper {X, 1 <_ k <_ n,n >_ 1} will denote rowwise independent random elements in such that for all n and k. (1.1) The major results of this paper show that n 1/ E X,t ---, 0 completely (1.2) where complete convergence is defined (as in Hsu and Robbins [1]) for each e > 0.
ErdSs [2] showed that for an array of i.i.d, random variables {X,k}, (1.3) holds if and only if E[XI] < o.Jain [3] obtained a uniform strong law of large numbers for sequences of i.i.d, random elements in separable Banach spaces of type 2 which would yield (1.2) with p 1 for an array of i.i.d, random elements {Xn} in a type 2 space.Woyczynski [4] showed that n/-.X, .--, 0completely (1.4)   for any sequence {X,} of independent random elements in a Banach space of type r, 1 _< p < r _< 2 with EX 0 for all n which is uniformly bounded by a random variable X satisfying EIX[P < oo.Recsll that an array {X,k} of random elements is said to be uniformly bounded by a random variable X if for all n and k and for every real number (1.5) Note that i.i.d, random elements are uniformly bounded by [[X, [[.Moricz, Hu, and Taylor [5] showed that Erd6s' result could be obtained by replacing the i.i.d, condition by the uniformly bounded condition (1.5).Taylor and Hu [6] obtained complete convergence in type r spaces, 1 < r _< 2 for uniformly bounded; rowwise independent random elements.
The results of this paper relaxes the assumption of uniformly bounded random elements in Taylor and Hu [6].Moreover, a major application of the main result of this paper is indicated for kernel density estimators where uniformly bounded random variables can not be asumed.
The following lemma from Woyczynski [4] will be used in obttdning the major result, Theorem 2.
LEMMA 1.Let 1 < r < 2 and q > 1.The following properties are equivalent: (i) is of type r (ii) There exists a C such that for MI independent random elements X1,... ,X, in with EXk O, mad EIIXII < oo, & --x E IIX.ll For vlues of p and r, 1 _< p < r _< 2, it follows that v > 2.Moreover, as p and r move dose to each other v increases without bound.However, for certain values of p strictly less than one, a value of v 1 is possible to obtain complete convergence.To see this letp= , r landa 0. It follows that v(-) v(3-1)= 2v > l implies that v > 1/2.However, the proof of Theorem 2 requires that v >_ 1.Thus, v 1 is the smallest moment necessary (given suitable conditions on p, r and a) to obtain complete convergence, via Theorem 2 REMARK 2. The condition sup EI[X,kI[" O(n') is somewhat stronger than (1.5) used by Taylor _<k_<and Hu [6].However, the bound in each row increases as n oo which is a substantia/ improvement in Theorem 4 of Taylor,/Ioricz and Hu [5].This substantil improvement will be illustrated in Exv.mple I.
An immediate corol/ary to Theorem 2 is obtained for i.i.d, random elements.
COROLLARY 3. Let {X,) be an array of i.i.d, random elements in a Banach space of type r such that EX11 0. Let EIIXll < o where ( ) > 1,0 < p < r _< 2.Then, The moment condition in Corollary 3 can be considerably smaller than the moment condition in Theorem 6 of Taylor and Hu [6], (see Remark 1) but in general will be much larger.