A renewal theorem for relatively stable variables

TL;DR

This paper derives an exact asymptotic form of the Green measure for relatively stable distributions, extending classical renewal theorems to a broader class of probability distributions on the real line.

Contribution

It provides a new renewal theorem for relatively stable variables, relaxing previous conditions and establishing precise asymptotics for the Green measure as the variable tends to infinity.

Findings

Established asymptotic behavior of the Green measure for relatively stable distributions.

Extended renewal theorems beyond the classical assumptions.

Provided conditions under which the limit of the scaled Green measure exists.

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.