Towards Nikishin's Theorem on the Almost Sure Convergence of Rearrangements of Functional Series

Necessary and sufficient conditions are found for the almost sure convergence of almost all simple rearrangements of a series of Banach space valued random variables. The results go back to Nikishin's well-known theorem on the existence of an almost surely convergent rearrangement of a numerical random series. An example is also given of a numerical random series with general term tending to zero almost surely such that this series converges in probability and any its rearrangement diverges almost surely.