On the laws of large numbers for double arrays of independent random elements in Banach spaces (Q741209)

The paper considers a double array \(\{V_{mn}, m \geq 1, n \geq 1\}\) of independent random elements on a probability space \((\Omega, \mathcal{F},P)\) and taking values in a real separable Banach space \(\chi\) with norm \(||\cdot||\). It provides conditions under which the strong law of large numbers (SLLN) and the weak law of large numbers (WLLN) for the double sum \(\sum_{i=1}^n \sum_{j=1}^m V_{ij}\) are equivalent. Such double sums differ substantially from the the partial sums \(\sum_{i=1}^n V_i,\; n \geq 1\), of a sequence of independent random elements \(\{V_n, n \geq 1\}\). The paper treats both the i.i.d case and the independent but not identically distributed case. The main results are applied to obtain Kolmogorov, Brunk-Chung and Marcinkiewicz-Zygmund type SLLNs for double sums in Rademacher type \(p\) \((1 \leq p \leq 2)\) Banach spaces.