Almost sure convergence of randomly weighted series

In this Note, we obtain criterion for the convergence almost everywhere of series of contractions (of an arbitrary L-2-space) with random weights. It is a continuation of a previous work [4] in which only convergence in operator norm was investigated. We find conditions ensuring the existence of universal sets on which these series are converging almost everywhere, whatsoever the contraction and f epsilon L-2- It is also a continuation of the papers [1] and [6] in which an analog problem concerning ergodic averages was considered, as well as the paper [7] studying a variant of the problem. The proofs of the results rely on uniform estimates of random polynomials recently obtained by the second named author and proved by means of metric entropy methods arising form the stochastic processes theory. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.