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

This article continues work by Alsmeyer and Hoefs (Markov Process Relat. Fields 7 (2001) 325–348) on random walks (Sn)n⩾0 whose increments Xn are (m+1)-block factors of the form ϕ(Yn−m,…,Yn) for i.i.d. random variables Y−m,Y−m+1,… taking values in an arbitrary measurable space (S,S). Defining Mn=(Yn−m,…,Yn) for n⩾0, which is a Harris ergodic Markov chain, the sequence (Mn,Sn)n⩾0 constitutes a Markov random walk with stationary drift μ=EFm+1X1 where F denotes the distribution of the Ynʹs. Suppose μ>0, let (σn)n⩾0 be the sequence of strictly ascending ladder epochs associated with (Mn,Sn)n⩾0 and let (Mσn,Sσn)n⩾0, (Mσn,σn)n⩾0 be the resulting Markov renewal processes whose common driving chain is again positive Harris recurrent. The Markov renewal measures associated with (Mn,Sn)n⩾0 and the former two sequences are denoted Uλ,Uλ> and Vλ>, respectively, where λ is an arbitrary initial distribution for (M0,S0). Given the basic sequence (Mn,Sn)n⩾0 is spread-out or 1-arithmetic with shift function 0, we provide convergence rate results for each of Uλ,Uλ> and Vλ> 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.