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

For a double array of independent random elements {V mn ,m . 1, n . 1} in a real separable Banach space, conditions are provided under which the weak and strong laws of large numbers for the double sums . mi=1 . nj=1 V ij , m . 1, n . 1 are equivalent. Both the identically distributed and the nonidentically distributed cases are treated. In the main theorems, no assumptions are made concerning the geometry of the underlying Banach space. These theorems are applied to obtain Kolmogorov, Brunk-Chung, and Marcinkiewicz-Zygmund type strong laws of large numbers for double sums in Rademacher type p (1 . p . 2) Banach spaces.