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Periodic regeneration

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  • Thorisson, Hermann

Abstract

We consider stochastic processes Z = (Zt)[0,[infinity]), on a general state space, having a certain periodic regeneration property: there is an increasing sequence of random times (Sn)[infinity]0 such that the post-Sn process is conditionally independent of S0,...,Sn given (Sn mod 1) and the conditional distribution does not depend on n. Our basic condition is that the distributions , have a common component that is absolutely continuous w.r.t. Lebesgue measure. Then Z has the following time-homogeneous regeneration property: there exists a discrete aperiodic renewal process T = (Tn)[infinity]0 such that the post-Tn process is independent of T0,...,Tn and its distribution does not depend on n; this yields weak ergodicity. Further, the Markov chain (Sn mod 1)[infinity]0 has an invariant distribution [pi][0,1) and it holds that Tn+1 - Tn has finite first moment if and only if m = [integral operator] m(Ps)[pi][0,1)(ds)

Suggested Citation

  • Thorisson, Hermann, 1985. "Periodic regeneration," Stochastic Processes and their Applications, Elsevier, vol. 20(1), pages 85-104, July.
    • Handle: RePEc:eee:spapps:v:20:y:1985:i:1:p:85-104
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