An arcsine law for Markov random walks

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:129

Article

Alsmeyer, Gerold¹  Buckmann, Fabian¹ 

[1] Univ Munster, Dept Math & Comp Sci, Inst Math Stochast, Orleans Ring 10, D-48149 Munster, Germany

关键词: Markov random walk;    Arcsine law;    Fluctuation theory;    Spitzer condition;    Spitzer-type formula;   

DOI  :  10.1016/j.spa.2018.02.014

来源: Elsevier

【 摘 要 】

The classic arcsine law for the number N-n(>) := n(-1)Sigma(n )(k=1)1({Sk > 0}) of positive terms, as n -> infinity 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 (Mn, Sn)(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 rho is an element of [0, 1] (the classic arcsine law if rho = 1/2) iff the Spitzer condition lim(n -> infinity) 1/n Sigma P-n(k=1)i(S-n > 0) = rho holds true for some and then all i is an element of S, where P-i := P(.vertical bar M-0 = i) for i is an element of S. It is also proved, under an extra assumption on the driving chain if 0 < rho < 1, that this condition is equivalent to the stronger variant lim(n -> infinity) P-i(S-n > 0) = rho. For an ordinary random walk, this was shown by Doney (1995) for 0 < p < 1 and by Bertoin and Doney (1997) for rho is an element of {0, 1}. (C) 2018 Elsevier B.V. All rights reserved.

【 授权许可】

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