On the Limiting Ratio of Current Age to Total Life for Null Recurrent Renewal Processes
arXiv: Probability(2015)
摘要
If the inter-arrival time distribution of a renewal process is regularly varying with index $alphainleft( 0,1right) $ (i.e. the inter-arrival times have infinite mean) and if $Aleft( tright) $ is the associated age process at time $t$. Then we show that if $Cleft( tright) $ is the length of the current cycle at time $t$, [ Aleft( tright) /Cleft( tright) Rightarrow U^{1/alpha}, ] where $U$ is $Uleft( 0,1right) $. This extends a classical result in renewal theory in the finite mean case which indicates that the limit is $Uleft( 0,1right) $.
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