A Counterexample in Renewal Theory

Abstract

The purpose of this note is to give a counterexample to the following statement. Let $Y_1, Y_2, \cdots$ be i.i.d rv with distribution function $F$ and $P\lbrack Y_1 \geqq 0\rbrack = 1$. For any set $A \subset \lbrack 0, \infty)$ let $U(A) = \sum^\infty_{k=0} F^{\ast k}(A)$ be the usual renewal measure. If $A \subset \lbrack 0, \infty)$ and $U(A) = +\infty$ then there is a renewal in $A$ almost surely.