Markov renewal theory for stationary ( m+1)-block factors: convergence rate results

Abstract

This article continues work by Alsmeyer and Hoefs (Markov Process Relat. Fields 7 (2001) 325–348) on random walks ( S n ) n⩾0 whose increments X n are ( m+1)-block factors of the form ϕ( Y n− m ,…, Y n ) for i.i.d. random variables Y − m , Y − m+1 ,… taking values in an arbitrary measurable space ( S, S) . Defining M n =( Y n− m ,…, Y n ) for n⩾0, which is a Harris ergodic Markov chain, the sequence ( M n , S n ) n⩾0 constitutes a Markov random walk with stationary drift μ= E F m+1 X 1 where F denotes the distribution of the Y n 's. Suppose μ>0, let ( σ n ) n⩾0 be the sequence of strictly ascending ladder epochs associated with ( M n , S n ) n⩾0 and let ( M σ n , S σ n ) n⩾0 , ( M σ n , σ n ) n⩾0 be the resulting Markov renewal processes whose common driving chain is again positive Harris recurrent. The Markov renewal measures associated with ( M n , S n ) n⩾0 and the former two sequences are denoted U λ , U λ > and V λ >, respectively, where λ is an arbitrary initial distribution for ( M 0, S 0). Given the basic sequence ( M n , S n ) n⩾0 is spread-out or 1-arithmetic with shift function 0, we provide convergence rate results for each of U λ , U λ > and V λ > under natural moment conditions. Proofs are based on a suitable reduction to standard renewal theory by finding an appropriate imbedded regeneration scheme and coupling. Considerable work is further spent on necessary moment results.