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