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Markov renewal theory for stationary (m+1)-block factors: convergence rate results

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  • Alsmeyer, Gerold
  • Hoefs, Volker

Abstract

This article continues work by Alsmeyer and Hoefs (Markov Process Relat. Fields 7 (2001) 325-348) on random walks (Sn)n[greater-or-equal, slanted]0 whose increments Xn are (m+1)-block factors of the form [phi](Yn-m,...,Yn) for i.i.d. random variables Y-m,Y-m+1,... taking values in an arbitrary measurable space . Defining Mn=(Yn-m,...,Yn) for n[greater-or-equal, slanted]0, which is a Harris ergodic Markov chain, the sequence (Mn,Sn)n[greater-or-equal, slanted]0 constitutes a Markov random walk with stationary drift [mu]=EFm+1X1 where F denotes the distribution of the Yn's. Suppose [mu]>0, let ([sigma]n)n[greater-or-equal, slanted]0 be the sequence of strictly ascending ladder epochs associated with (Mn,Sn)n[greater-or-equal, slanted]0 and let (M[sigma]n,S[sigma]n)n[greater-or-equal, slanted]0, (M[sigma]n,[sigma]n)n[greater-or-equal, slanted]0 be the resulting Markov renewal processes whose common driving chain is again positive Harris recurrent. The Markov renewal measures associated with (Mn,Sn)n[greater-or-equal, slanted]0 and the former two sequences are denoted U[lambda],U[lambda]> and V[lambda]>, respectively, where [lambda] is an arbitrary initial distribution for (M0,S0). Given the basic sequence (Mn,Sn)n[greater-or-equal, slanted]0 is spread-out or 1-arithmetic with shift function 0, we provide convergence rate results for each of U[lambda],U[lambda]> and V[lambda]> under natural moment conditions. Proofs are based on a suitable reduction to standard renewal theory by finding an appropriate imbedded regeneration scheme and coupling. Considerable work is further spent on necessary moment results.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:98:y:2002:i:1:p:77-112
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    References listed on IDEAS

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    1. Alsmeyer, Gerold, 1994. "On the Markov renewal theorem," Stochastic Processes and their Applications, Elsevier, vol. 50(1), pages 37-56, March.
    2. Niemi, S. & Nummelin, E., 1986. "On non-singular renewal kernels with an application to a semigroup of transition kernels," Stochastic Processes and their Applications, Elsevier, vol. 22(2), pages 177-202, July.
    3. 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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