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

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) $.