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On the Markov renewal theorem

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  • Alsmeyer, Gerold

Abstract

Let (S, £) be a measurable space with countably generated [sigma]-field £ and (Mn, Xn)n[greater-or-equal, slanted]0 a Markov chain with state space S x and transition kernel :S x ( [circle times operator] )-->[0, 1]. Then (Mn,Sn)n[greater-or-equal, slanted]0, where Sn = X0+...+Xn for n[greater-or-equal, slanted]0, is called the associated Markov random walk. Markov renewal theory deals with the asymptotic behavior of suitable functionals of (Mn,Sn)n[greater-or-equal, slanted]0 like the Markov renewal measure [Sigma]n[greater-or-equal, slanted]0P((Mn,Sn)[epsilon]Ax (t+B)) as t-->[infinity] where A[epsilon] and B denotes a Borel subset of . It is shown that the Markov renewal theorem as well as a related ergodic theorem for semi-Markov processes hold true if only Harris recurrence of (Mn)n[greater-or-equal, slanted]0 is assumed. This was proved by purely analytical methods by Shurenkov [15] in the one-sided case where (x,Sx[0,[infinity])) = 1 for all x[epsilon]S. Our proof uses probabilistic arguments, notably the construction of regeneration epochs for (Mn)n[greater-or-equal, slanted]0 such that (Mn,Xn)n[greater-or-equal, slanted]0 is at least nearly regenerative and an extension of Blackwell's renewal theorem to certain random walks with stationary, 1-dependent increments.

Suggested Citation

  • Alsmeyer, Gerold, 1994. "On the Markov renewal theorem," Stochastic Processes and their Applications, Elsevier, vol. 50(1), pages 37-56, March.
    • Handle: RePEc:eee:spapps:v:50:y:1994:i:1:p:37-56
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    Cited by:

    1. Masakiyo Miyazawa, 2011. "Light tail asymptotics in multidimensional reflecting processes for queueing networks," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(2), pages 233-299, December.
    2. Fuh, Cheng-Der, 2021. "Asymptotic behavior for Markovian iterated function systems," Stochastic Processes and their Applications, Elsevier, vol. 138(C), pages 186-211.
    3. Hansen, Niels Richard & Jensen, Anders Tolver, 2005. "The extremal behaviour over regenerative cycles for Markov additive processes with heavy tails," Stochastic Processes and their Applications, Elsevier, vol. 115(4), pages 579-591, April.
    4. Basrak, Bojan & Conroy, Michael & Olvera-Cravioto, Mariana & Palmowski, Zbigniew, 2022. "Importance sampling for maxima on trees," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 139-179.
    5. Gobet, Emmanuel & Menozzi, Stéphane, 2010. "Stopped diffusion processes: Boundary corrections and overshoot," Stochastic Processes and their Applications, Elsevier, vol. 120(2), pages 130-162, February.
    6. Haas, Bénédicte & Stephenson, Robin, 2018. "Bivariate Markov chains converging to Lamperti transform Markov additive processes," Stochastic Processes and their Applications, Elsevier, vol. 128(10), pages 3558-3605.
    7. Gobet, Emmanuel & Menozzi, Stéphane, 2004. "Exact approximation rate of killed hypoelliptic diffusions using the discrete Euler scheme," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 201-223, August.
    8. Dmitry Korshunov, 2008. "The Key Renewal Theorem for a Transient Markov Chain," Journal of Theoretical Probability, Springer, vol. 21(1), pages 234-245, March.
    9. Alsmeyer, Gerold & Hoefs, Volker, 2002. "Markov renewal theory for stationary (m+1)-block factors: convergence rate results," Stochastic Processes and their Applications, Elsevier, vol. 98(1), pages 77-112, March.
    10. Fuh, Cheng-Der & Zhang, Cun-Hui, 2000. "Poisson equation, moment inequalities and quick convergence for Markov random walks," Stochastic Processes and their Applications, Elsevier, vol. 87(1), pages 53-67, May.
    11. Alsmeyer, Gerold, 1996. "Superposed continuous renewal processes A Markov renewal approach," Stochastic Processes and their Applications, Elsevier, vol. 61(2), pages 311-322, February.

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