Abstract:

Let 1 Pn=1 Xn be a series of independent random variables with at least one non-degenerate Xn, and let Fn be the distribution function of its partial sums Sn =n Pk=1 Xk. Motivated by Hildebrand’s work in [1], the authors investigate the a.s. convergence of 1 Pn=1 Xn under a hypothesis that 1
Pn=1 (Xn, cn) = 1 whenever 1 Pn=1 cn diverges, where the notation (X, c) denotes the L´evy distance between the random variable X and the
constant c. The principal result of this paper shows that the hypothesis is the condition under which the convergence of Fn(x0) with the limit value 0 < L0 < 1, together with the essential convergence of 1 Pn=1 Xn, is both sufficient and necessary in order for the series 1 Pn=1 Xn to a.s. converge. oreover, if the essential convergence of 1 Pn=1 Xn is strengthened to lisup n!1 P(|Sn| < K) = 1 for some K > 0, the hypothesis is already equivalent to the a.s. convergence of 1 Pn=1 Xn. Here they have not only founded a very general limit theorem, but improved the related result in Hildebrand[1] as well.

Key words: Sums of independent random variabies, essential convergence, limit distrib-ution, Levy distance, three series theorem