Abstract. Let $p \in(0, \infty )$ be a constant and let $\{\xi_n\} \subset L^p(\Omega, {\cal F}, {\msbm P})$ be a sequence of random variables. For any integers $m, n \ge0$, denote $S_{m, n} = \sum_{k=m}^{m + n-1} \xi_k$. It is proved that, if there exist a nondecreasing function $\varphi\colon {\msbm R}_+\to{\msbm R}_+$ (which satisfies a mild regularity condition) and an appropriately chosen integer $a\ge2$ such that $$\sum_{n=0}^\infty\sup_{k \ge0} {\msbm E}\left|\frac{S_{k, a^n}} {\varphi(a^n)}\right|^p < \infty,\ \hbox{ then }\ \lim_{n \to\infty } {S_{0, n}\over\varphi (n)} = 0 \ \hbox{ a.s.} $$ This extends Theorem 1 in Chobanyan, Levental and Salehi [chobanyan-l-s] and can be applied conveniently to a wide class of self-similar processes with stationary increments including stable processes.

AMS Subject Classification (1991): 60F15

Keyword(s): strong law of large numbers, moment inequality, self-similar processes, stable processes

Received May 1, 2009, and in revised form December 17, 2009. (Registered under 64/2009.)