Necessary and sufficient conditions for complete convergence of double weighted sums of pairwise independent identically distributed random elements in Banach spaces

This paper provides necessary and sufficient conditions for complete convergence of double weighted sums of pairwise independent identically distributed random elements in Banach spaces. The main theorem extends Theorem 1.2 in [1] to the double weighted sum setting. The sharpness of the main result is illustrated by showing that the main theorem can fail if we replace the identical distribution condition by a slightly weaker condition, even when the random elements are independent and uniformly almost surely bounded.