On the limiting behavior of tail series

The rate of convergence for an almost certainly convergent series Sn=∑j=1nXj of random variables is studied in this paper. More specifically, when Sn converges almost certainly to a random variable S, the tail series Tn≡S-Sn-1=∑j=n∞Xj is a well-defined sequence of random variables with Tn→0 almost certainly. Let {bn,n⩾1} be a sequence of positive constants. The main result provides for independent {Xn,n⩾1} conditions for each of the one-sided implicationslimsupn→∞Xnbn=∞almost certainly⇒limsupn→∞Tnbn=∞almost certainlyandliminfn→∞Xnbn=-∞almost certainly⇒liminfn→∞Tnbn=-∞almost certainlyto hold. Furthermore, a tail series strong law of large numbers (SLLN) Tn/bn→0 almost certainly is proved without assuming the {Xn,n⩾1} are independent where bn→0 very rapidly. Illustrative examples are provided concerning various aspects of the results, the last of which shows that they are sharp.