On the Limiting Ratio of Current Age to Total Life for Null Recurrent Renewal Processes

TL;DR

This paper investigates the asymptotic behavior of the ratio of current age to cycle length in null recurrent renewal processes with infinite mean inter-arrival times, extending classical finite-mean results.

Contribution

It extends classical renewal theory results to cases with infinite mean inter-arrival times, showing the limiting distribution of the age-to-cycle ratio as a power of a uniform variable.

Findings

The ratio converges in distribution to U^{1/α} where U is uniform(0,1).

The result generalizes finite mean renewal process behavior to infinite mean cases.

Provides a new limit theorem for null recurrent renewal processes.

Abstract

If the inter-arrival time distribution of a renewal process is regularly varying with index $\alpha\in\left( 0,1\right) $ (i.e. the inter-arrival times have infinite mean) and if $A\left( t\right) $ is the associated age process at time $t$. Then we show that if $C\left( t\right) $ is the length of the current cycle at time $t$, \[ A\left( t\right) /C\left( t\right) \Rightarrow U^{1/\alpha}, \] where $U$ is $U\left( 0,1\right) $. This extends a classical result in renewal theory in the finite mean case which indicates that the limit is $U\left( 0,1\right) $.