An Arcsine Law for Markov Random Walks

Autor: Gerold Alsmeyer, Fabian Buckmann
Rok vydání: 2017
Předmět:
DOI: 10.48550/arxiv.1703.00316
Popis: 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 } .
Databáze: OpenAIRE
Externí odkaz: