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December, 1978 On Semi-Markov and Semiregenerative Processes II
David McDonald
Ann. Probab. 6(6): 995-1014 (December, 1978). DOI: 10.1214/aop/1176995389


An ergodic theorem is given for the age process $(I(t), Z(t))$ associated with a (possibly transient) semi-Markov chain $(I_n, X_n)^\infty_{n=0}$ whose sojourn times are not exclusively integer valued. Asymptotically the Markov part $(I(t):$ the state occupied at time $t)$ and the renewal part $(Z(t):$ the age in $I(t)$ at time $t)$ split into independent parts. This yields the following ergodic result for a semiregenerative process $V_t$ with embedded semi-Markov chain $(I_n, X_n)^\infty_{n=0}$: $$\lim_{t\rightarrow \infty}\big| \operatorname{Prob}\{V_t \in A\} - \int_\pi\frac{A_\pi}{\mu_\pi} \operatorname{Prob}\{I(t) = d\pi\}\big| = 0$$ where $\pi$ is in the state space of $I_n, \mu_\pi$ is the mean sojourn time in $\pi$ and $A_\pi$ is the mean time $V_t$ is in a set $A$ during a sojourn in $\pi$.


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David McDonald. "On Semi-Markov and Semiregenerative Processes II." Ann. Probab. 6 (6) 995 - 1014, December, 1978.


Published: December, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0395.60081
MathSciNet: MR512416
Digital Object Identifier: 10.1214/aop/1176995389

Primary: 60K15
Secondary: 60F99 , 60J25 , 60K05

Keywords: Ergodic , nonrecurrent , semi-Markov process , semi-regenerative process

Rights: Copyright © 1978 Institute of Mathematical Statistics

Vol.6 • No. 6 • December, 1978
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