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An arcsine law for Markov random walks.

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Title: An arcsine law for Markov random walks.
Authors: Alsmeyer, Gerold1 gerolda@math.uni-muenster.de, Buckmann, Fabian1 f_buck01@uni-muenster.de
Source: Stochastic Processes & Their Applications. Jan2019, Vol. 129 Issue 1, p223-239. 17p.
Subjects: Arcsine function, Random walks, Markov processes, Markov chain Monte Carlo, Discrete probability theory
Abstract: Abstract The classic arcsine law for the number N n > ≔ n − 1 ∑ k = 1 n 1 { S k > 0 } of positive terms, as n → ∞ , in an ordinary random walk (S n) n ≥ 0 is extended to the case when this random walk is governed by a positive recurrent Markov chain (M n) n ≥ 0 on a countable state space S , that is, for a Markov random walk (M n , S n) n ≥ 0 with positive recurrent discrete driving chain. More precisely, it is shown that n − 1 N n > 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 lim n → ∞ 1 n ∑ k = 1 n P i (S n > 0) = ρ holds true for some and then all i ∈ S , where P i ≔ P (⋅ | M 0 = 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 lim n → ∞ P i (S n > 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 }. [ABSTRACT FROM AUTHOR]
Copyright of Stochastic Processes & Their Applications is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Abstract The classic arcsine law for the number N n &gt; ≔ n − 1 ∑ k = 1 n 1 { S k &gt; 0 } of positive terms, as n → ∞ , in an ordinary random walk (S n) n ≥ 0 is extended to the case when this random walk is governed by a positive recurrent Markov chain (M n) n ≥ 0 on a countable state space S , that is, for a Markov random walk (M n , S n) n ≥ 0 with positive recurrent discrete driving chain. More precisely, it is shown that n − 1 N n &gt; 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 lim n → ∞ 1 n ∑ k = 1 n P i (S n &gt; 0) = ρ holds true for some and then all i ∈ S , where P i ≔ P (⋅ | M 0 = i) for i ∈ S. It is also proved, under an extra assumption on the driving chain if 0 &lt; ρ &lt; 1 , that this condition is equivalent to the stronger variant lim n → ∞ P i (S n &gt; 0) = ρ. For an ordinary random walk, this was shown by Doney (1995) for 0 &lt; ρ &lt; 1 and by Bertoin and Doney (1997) for ρ ∈ { 0 , 1 }. [ABSTRACT FROM AUTHOR]
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  Data: &lt;i&gt;Copyright of Stochastic Processes &amp; Their Applications is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder&#39;s express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.&lt;/i&gt; (Copyright applies to all Abstracts.)
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        Value: 10.1016/j.spa.2018.02.014
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        Text: English
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        PageCount: 17
        StartPage: 223
    Subjects:
      – SubjectFull: Arcsine function
        Type: general
      – SubjectFull: Random walks
        Type: general
      – SubjectFull: Markov processes
        Type: general
      – SubjectFull: Markov chain Monte Carlo
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      – SubjectFull: Discrete probability theory
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      – TitleFull: An arcsine law for Markov random walks.
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              Text: Jan2019
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              Y: 2019
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