A strong law for $B$-valued arrays

Abstract:

Let $(B,\left \| \bullet \right \|)$ be a real separable Banach space and $\{ {X_{n,k}};n \geq 1,1 \leq k \leq n\}$ a triangular array of iid B-valued random variables. Set $S(n) = \sum \nolimits _{k = 1}^n {{X_{n,k}},n \geq 1}$, and ${\operatorname {Log}} t = \log \max \{ e,t\} ,t \in \Re$. In this paper, we characterize the limit behavior of $S(n)/\sqrt {2n {\operatorname {Log}} n} ,n \geq 1$. As an application of our result, we resolve an open problem posed by Hu and Weber (1992). The case of row-wise independent arrays is also dealt with.