EXACT ASYMPTOTIC BEHAVIOUR IN A RENEWAL THEOREM FOR CONVOLUTION EQUIVALENT DISTRIBUTIONS WITH EXPONENTIAL TAILS*

Abstract

We study the exact asymptotic behaviour of some special generalized renewal measures on the whole line $\mathbf{R}$ and, in particular, that of the ordinary renewal measure on $\mathbf{R}$. When the underlying distribution $F$ is concentrated on $\left[0,\infty \right)$, these measures are closely related to the higher renewal moments $EN{(t)}^n$, where $N\left(t\right)$ is the number of renewals up to time $t$. The tail of $F$ is assumed to possess the tail behaviour of a distribution from the class $S\left(\gamma \right)$ for an arbitrary .