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April, 1988 Backward Limits
Hermann Thorisson
Ann. Probab. 16(2): 914-924 (April, 1988). DOI: 10.1214/aop/1176991796


We consider a time-inhomogeneous regenerative process starting from regeneration at time $s$ and prove, under regularity conditions on the regeneration times, that the distribution of the process in a fixed time interval $\lbrack t, \infty)$ stabilizes as the starting time $s$ tends backward to $-\infty$ (the convergence considered here is in the sense of total variation). This implies the existence of a two-sided time-inhomogeneous process "starting from regeneration at $-\infty$." We also show that if a time-inhomogeneous regenerative process admits a limit law in the traditional forward sense, then it is asymptotically time-homogeneous; thus the backward approach widely extends the class of processes admitting a limit law.


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Hermann Thorisson. "Backward Limits." Ann. Probab. 16 (2) 914 - 924, April, 1988.


Published: April, 1988
First available in Project Euclid: 19 April 2007

zbMATH: 0643.60033
MathSciNet: MR929087
Digital Object Identifier: 10.1214/aop/1176991796

Primary: 60G07
Secondary: 60G20 , 60J99

Keywords: Backward limits , inhomogeneous Markov process , inhomogeneous regeneration , Regenerative process , two-sided process

Rights: Copyright © 1988 Institute of Mathematical Statistics


Vol.16 • No. 2 • April, 1988
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