On the Limiting Ratio of Current Age to Total Life 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 $t$, \[ A\left( t\right) /C\left( t\right) \Rightarrow U^{1/α}, \] 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 $U\left(0,1\right)$.

arXiv comment3 pages