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

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 interarrival times have infinite mean) are solved. In particular, when the survival function for the inter-arrival times satisfies (F) over bar (t) similar to t(-alpha)L(t) as t -> infinity with L slowly varying and 0 <= alpha <= 1, we prove that the limit distribution corresponds to that of U-1/alpha, where U is uniformly distributed on (0, 1), with the limit distribution taken to be degenerate at 0 when alpha = 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. (C) 2020 Elsevier B.V. All rights reserved.