On the ratio of current age to total life for null recurrent renewal processes

Abstract

A number of open problems associated with determining the limit distribution of the ratio of current age to total life for a null recurrent renewal process (i.e. where inter-arrival times have infinite mean) are solved. In particular, when the survival function for the inter-arrival times satisfies F̄(t)∼t−αL(t) as t→∞ with L slowly varying and 0≤α≤1, we prove that the limit distribution corresponds to that of U1∕α, where U is uniformly distributed on (0,1), with the limit distribution taken to be degenerate at 0 when α=0. By using direct methods instead of appealing to strong renewal theorems, we are able to prove this result without regard to whether the inter-arrival time distribution is latticed or not, and without extraneous constraints on the renewal function.