Strong Renewal Theorems with Infinite Mean

Abstract

Let F be a nonarithmetic probability distribution on $(0,\infty )$ and suppose $1 - F(t)$ is regularly varying at $\infty$ with exponent $\alpha ,0 < \alpha \leqq 1$. Let $U(t) = \Sigma {F^{{n^ \ast }}}(t)$ be the renewal function. In this paper we first derive various asymptotic expressions for the quantity $U(t + h) - U(t)$ as $t \to \infty ,h > 0$ fixed. Next we derive asymptotic relations for the convolution ${U^ \ast }z(t),t \to \infty$, for a large class of integrable functions z. All of these asymptotic relations are expressed in terms of the truncated mean function $m(t) = \smallint _0^t[1 - F(x)]dx$, t large, and appear as the natural extension of the classical strong renewal theorem for distributions with finite mean. Finally in the last sections of the paper we apply the special case $\alpha = 1$ to derive some limit theorems for the distributions of certain waiting times associated with a renewal process.