An arcsine law for Markov random walks

Abstract

The classic arcsine law for the number N > n := n −1 n k=1 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 n k=1 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 [7] for 0 < ρ < 1 and by Bertoin and Doney [3] for ρ ∈ {0, 1}.

Fluctuation Theory for Markov Random Walks

“…We also write f (x) g(x) and f (x) g(x) as shorthand for the left and the right relation in (8), respectively. Finally, given two expressions A, B (series or integrals), A ≍ B, A B and A B will be used if, for some…”

“…XI] and [6] may be consulted for more recent treatments of some aspects of the theory including the discrete Markov renewal theorem, the dual process, and Wiener-Hopf factorization, for the latter see also [12,44,24]. The ladder variables and the associated ladder chain of a MRW have been studied in [4,7], see also Section 4 for further information, an arcsine law for the number of positive sums is derived in [8], and the topological recurrence of (S n ) n≥0 in the case when E π X 1 = 0 is shown in [5]. For conditional Markov renewal theorems in the case when S is countable, we mention an article by Lalley [34].…”