On convergence of series of random variables with applications to martingale convergence and to convergence of series with orthogonal summands

Abstract

The rate of convergence for an almost surely convergent series of random variables {Xn, n ≥ 1} is studied in this paper. More specifically, if Sn converges almost surely to a random variable S then the tail series is a well-defined sequence of random variable with Tn → 0 almost surely. The main result provides conditions for each of to hold for given numerical sequences . As special cases, new results are obtained when {Xn, n ≥ 1} is a martingale difference sequence and when the Xn 's are of the form , where {an, n ≥ 1} is a sequence of constants and {Yn, n ≥ 1} is an orthogonal system of exchangeable random variables. An application to the Polya urn scheme is considered. The current work generalizes and simplifies some of the recent results of Nam and Rosalsky [15]