Bibliographic Details
| 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] |
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| Database: |
Engineering Source |
