Ladder epochs and ladder chain of a Markov random walk with discrete driving chain. (1st February 2019)

• Record Type: Journal Article
• Authors: Alsmeyer, Gerold
• Editors: Asmussen, S.; Klebaner, F.; Nerman, O.; Vatutin, V.
• Is Part Of: Advances in applied probability. Volume 50:Number A(2018)
• Page Start: 31
• Page End: 46
• Publication Date: 2019-02-01
• Subjects: Markov‐modulated sequence, -- Markov random walk, -- discrete Markov chain, -- ladder variable, -- ladder chain, -- excursion chain, -- stationary distribution, -- coupling, -- Palm‐duality, -- Wiener‒Hopf factorization; Primary 60J10, -- Secondary 60K15; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2018.68
• Languages: English
• ISSNs: 0001-8678

Abstract

Abstract: Let ( M n, S n ) n ≥0 be a Markov random walk with positive recurrent driving chain ( M n ) n ≥0 on the countable state space 𝒮 with stationary distribution π. Suppose also that lim sup n →∞ S n =∞ almost surely, so that the walk has almost-sure finite strictly ascending ladder epochs σ n > . Recurrence properties of the ladder chain ( M σ n > ) n ≥0 and a closely related excursion chain are studied. We give a necessary and sufficient condition for the recurrence of ( M σ n > ) n ≥0 and further show that this chain is positive recurrent with stationary distribution π > and 𝔼π > σ1 > <∞ if and only if an associated Markov random walk (𝑀̂ n, 𝑆̂ n ) n ≥0, obtained by time reversal and called the dual of ( M n, S n ) n ≥0, is positive divergent, i.e. 𝑆̂ n →∞ almost surely. Simple expressions for π > are also provided. Our arguments make use of coupling, Palm duality theory, and Wiener‒Hopf factorization for Markov random walks with discrete driving chain.