Almost sure convergence of weighted series of contractions

In this paper we consider the almost sure convergence of a series of contractions (of an arbitrary Hilbert space) with random weights. The paper is a continuation of a previous work [PSW], in which only convergence in operator norm was investigated. We obtain conditions ensuring the existence of universal sets on which these series are converging almost everywhere, for any contraction. The paper is also a continuation of the paper [SW], in which an analogous problem concerning ergodic averages was considered, as well as the paper [S], which deals with a variant of the problem. The proofs of our results rely on uniform estimates of random polynomials which were established in a recent paper by the second author and proved by means of metric entropy methods.