Almost sure convergence for double arrays of block-wiseM-dependent random elements in Banach spaces

Abstract

For a double array of blockwise M-dependent random elements {Vmn : m 1, n 1} taking values in a real separable Rademacher type p (1 p 2) Banach space, we provide conditions to obtain the almost sure convergence for double sums , m 1, n 1. The paper treats two cases: (i) {Vmn : m 1, n 1} is block-wise M-dependent with EVmn 0, m, n 1, and (ii) {Vmn : m 1, n 1} is block-wise p-orthogonal. The conditions for case (i) are shown to provide exact characterizations of Rademacher type p and stable type p Banach spaces. Examples are given showing that the conditions cannot be removed or weakened. It is also demonstrated that some of the well-known theorems in the literature are special cases of our results.