Bull. Korean Math. Soc. 38 (2001), No. 4, pp. 763–772 ON CONVERGENCE OF SERIES OF INDEPENDENT RANDOM VARIABLES Soo Hak Sung and Andrei I. Volodin Abstract. The rate of convergence for an almost surely conver- gent series S n = ∑ n i=1 X i of independent random variables is stud- ied in this paper. More specifically, when S n converges almost surely to a random variable S, the tail series T n ≡ S - S n-1 = ∑ ∞ i=n X i is a well-defined sequence of random variables with T n → 0 almost surely. Conditions are provided so that for a given posi- tive sequence {b n ,n ≥ 1}, the limit law sup k≥n |T k |/b n P → 0 holds. This result generalizes a result of Nam and Rosalsky [4]. 1. Introduction Throughout this paper, {X n ,n ≥ 1} is a sequence of independent ran- dom variables defined on a probability space (Ω, F ,P ). As usual, their partial sums will be denoted by S n = ∑ n i=1 X i ,n ≥ 1. If S n converges almost surely (a.s.) to a random variable S, then (set S 0 = 0) T n ≡ S - S n−1 = ∞  i=n X i ,n ≥ 1 is a well-defined sequence of random variables (referred to as the tail series) with (1.1) T n → 0 a.s. Received April 6, 2001. 2000 Mathematics Subject Classification: 60F05, 60F15. Key words and phrases: convergence in probability, tail series, independent ran- dom variables, weak law of large numbers, almost sure convergence. The first author was supported by Korea Research Foundation Grant(KRF-2000- 015-DP0049).