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

Jose Blanchet; Peter Glynn; Hermann Thorisson

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arXiv · 1503.08374

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

Jose Blanchet·Peter Glynn·Hermann Thorisson

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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) $.

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Jose Blanchet, Peter Glynn, Hermann Thorisson. 2015-03-29. On the Limiting Ratio of Current Age to Total Life for Null Recurrent Renewal Processes. https://arxiv.org/abs/1503.08374

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