On Infinitely Divisible Laws and a Renewal Theorem for Non-Negative Random Variables

Abstract

Let $\{X_n\}$ be an infinite sequence of independent non-negative random variables such that, for some regularly varying non-decreasing function $\lambda(n)$, with exponent $1/\beta, 0 < \beta < \infty$, as $n \rightarrow \infty$, $P\{X_1 + \cdots + X_n/\lambda(n) \leqq x\} \rightarrow K(x)$ at all continuity points of some d.f. $K(x)$. Let $\Lambda(x)$ be the function inverse to $\lambda(n)$, let $R(x)$ be any other regularly varying function of exponent $\alpha > 0$. Then, if $N(x)$ is the maximum $k$ for which $X_1 + \cdots + X_k \leqq x$, it is proved that, as $x \rightarrow \infty$, $\varepsilon R(N(x)) \sim I(\alpha\beta)R(\Lambda(x))$ where $I(\alpha\beta) = \int^\infty_0 u^{\alpha\beta} dK(u)$ and this latter integral may diverge.