A strong renewal theorem for relatively stable variables

Kohei Uchiyama

Published 2019-11-25Version 1

Let $X$ be a non-negative random variable that is relatively stable, or what amounts to the same, $\int_0^x P[X>t]dt \sim \ell(x)$ for a slowly varying function $\ell$. We show that if $X$ is not arithmetic, the renewal function $U(x)$ associated with $X$ satisfies $\lim_{x\to\infty}[U(x+h)-U(x)]\ell(x) = h$ for $h>0$. An obvious analogue also holds for the arithmetic variable. The extension to not necessarily non-negative $X$ is obtained.