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

Let {X_(nk)} be an array of rowwise independent random elements in a separable Banach space of type p+δ with EX_(nk)=0 for all k, n. The complete convergence (and hence almost sure convergence) of n^(-1/p)∑_(k=1)^n X_(nk) to 0, 1≤p＜2, is obtained when {X_(nk)} are uniformly bounded by a random variable X with E|X|＆(2p)＜∞. When the array{X_(nk)} consists of i.i.d, random elements, then it is shown that n^(-1/p)∑_(k=1)^n X_(nk) converges completely to 0 if and only if E||‖X_(11)‖^(2p)＜∞.