A Renewal Theorem of Blackwell Type

Abstract

Suppose $\{X_1, X_2, \cdots\}$ are i.i.d. random variables with finite mean $0 < E(X_1) < \infty$. If $S_n$ stands for the $n$th partial sum, and $\{a(n)\}_n$ is a sequence of nonnegative numbers, then $G(x) = \sum^\infty_{n = 0} a(n)P\{S_n \leq x\}$ is a generalized renewal measure. We investigate the behaviour of $G(x + h) - G(x)$ as $x \rightarrow \infty$ for $\{a(n)\}_n$ regularly varying.