On the Weak Limiting Behavior of Almost Surely Convergent Row Sums from Infinite Arrays of Rowwise Independent Random Elements in Banach Spaces

Abstract

For an array {Vnk,k≥1,n≥1} of rowwise independent random elements in a real separable Banach space \(X\) with almost surely convergent row sums \(S_n = \sum {_{k = 1}^\infty {\text{ }}V_{nk} ,n \geqslant 1} \), we provide criteria for Sn−An to be stochastically bounded or for the weak law of large numbers \(S_n - A_n \xrightarrow{P}0\) to hold where {An,n≥1} is a (nonrandom) sequence in \(X\).