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An arcsine law for Markov random walks

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  • Alsmeyer, Gerold
  • Buckmann, Fabian

Abstract

The classic arcsine law for the number Nn>≔n−1∑k=1n1{Sk>0} of positive terms, as n→∞, in an ordinary random walk (Sn)n≥0 is extended to the case when this random walk is governed by a positive recurrent Markov chain (Mn)n≥0 on a countable state space S, that is, for a Markov random walk (Mn,Sn)n≥0 with positive recurrent discrete driving chain. More precisely, it is shown that n−1Nn> converges in distribution to a generalized arcsine law with parameter ρ∈[0,1] (the classic arcsine law if ρ=1∕2) iff the Spitzer condition limn→∞1n∑k=1nPi(Sn>0)=ρholds true for some and then all i∈S, where Pi≔P(⋅|M0=i) for i∈S. It is also proved, under an extra assumption on the driving chain if 0<ρ<1, that this condition is equivalent to the stronger variant limn→∞Pi(Sn>0)=ρ.For an ordinary random walk, this was shown by Doney (1995) for 0<ρ<1 and by Bertoin and Doney (1997) for ρ∈{0,1}.

Suggested Citation

  • Alsmeyer, Gerold & Buckmann, Fabian, 2019. "An arcsine law for Markov random walks," Stochastic Processes and their Applications, Elsevier, vol. 129(1), pages 223-239.
  • Handle: RePEc:eee:spapps:v:129:y:2019:i:1:p:223-239
    DOI: 10.1016/j.spa.2018.02.014
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    References listed on IDEAS

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    1. Woodroofe, Michael, 1992. "A central limit theorem for functions of a Markov chain with applications to shifts," Stochastic Processes and their Applications, Elsevier, vol. 41(1), pages 33-44, May.
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    Cited by:

    1. Gerold Alsmeyer & Chiranjib Mukherjee, 2023. "On Null-Homology and Stationary Sequences," Journal of Theoretical Probability, Springer, vol. 36(1), pages 1-25, March.

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