Teugels' Renewal Theorem and Stable Laws

Abstract

Let $\{S_n\}, n = 1,2, \cdots$ denote the partial sums of a sequence of independent, identically distributed nonnegative random variables with common distribution function $F$ having finite mean $\mu$, and let $H(t) = \sum^\infty_{n=1} P(S_n \leqq t)$. Further, let $F$ be nonarithmetic. It is shown in this paper that as $t \rightarrow \infty H(t) - t/\mu$ is regularly varying if and only if $F$ belongs to the domain of attraction of a stable law with exponent $\alpha, 1 < \alpha \leqq 2$.