Marcinkiewicz-Zygmund strong laws of large numbers for two-dimensional arrays of Banach space valued random variables (Q748999)

Let \(1\leq p,q<2\) and B a Banach space in which \(1_{p\vee q}\) is not finitely representable. Let \(\{X_{mn}\), \(m,n=1,2,...\}\) be a two- dimensional array of B-valued, independent mean zero random variables with tail probabilities uniformly bounded by tail probabilities of a real random variable X such that \(E | X|^ q<\infty\) if \(1\leq p<q<2\), \(E | X|^ p\log^+| X| <\infty\) if \(1<p=q<2\), and \(E | X| (\log^+| X|)^ 2<\infty\) if \(p=q=1.\) The author shows that the double series \(\sum_{m,n}X_{mn}/m^{1/q}n^{1/p}\) converges a.s. in norm and \(m^{-1/q}n^{-1/p}\sum_{i\leq m}\sum_{j\leq n}X_{ij}\) converges a.s. to 0. This extends to array form a similar result of \textit{W. A. Woyczyński} [Probab. Math. Stat. 1, 117-131 (1980; Zbl 0502.60006)] involving sequences \(\{X_ n\), \(n\geq 1\}\).