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.

17 pages, to appear in Bulletin of LMS