A renewal theorem for relatively stable variables

Abstract

Let $F\{dx\}$ be a relatively stable probability distribution on the whole real line and $S_n$ the random walk started at the origin with step distribution $F$. We obtain an exact asymptotic form of the Green measure $U\{x+dy\}= \sum_{n=0}^\infty P[S_n-x \in dy]$ as $x\to \infty$ when $S_n$ is transient and $S_n\to \infty$ in probability. If $F$ is concentrated on $[0,\infty)$, it is relatively stable if and only if $\ell(x) :=\int_0^x F\{(t,\infty)\}dt$ is slowly varying at infinity; our result entails that if $F$ is non-arithmetic and relatively stable, then $\lim_{x\to\infty}\, \ell(x)U\{[x, x+h)\} = h$ for each $h>0$. This surpasses the known result due to Erickson \cite{Ec}, the latter assuming the stronger condition that $xF\{(x,\infty)\}$ is slowly varying. An obvious analog also holds for arithmetic variables.