Markov Renewal Theory for Stationary m-Block Factors

G. Alsmeyer, V. Hoefs

2001, v.7, №2, 325-348

ABSTRACT

This article considers random walks $\SN$ whose increments $X_{n}$ are $(m+1)$-block factors of the form $\varphi(Y_{n-m},\dots,Y_{n})$ for i.i.d. random variables $Y_{-m},Y_{-m+1},\dots$ taking values in an arbitrary measurable space $(\cal S,\mathfrak S)$. Providing $E X_{1}>0$ and by further introducing the Markov chain $M_{n}=(Y_{n-m},\dots$, $Y_{n}),$ $n\ge 0$, they can be perfectly analyzed within the framework of Markov renewal theory. Our approach leads to new results for the associated sequence of ladder variables as well as for the first passage times $\tau(t)=\inf\{n\ge 1: S_{n}>t\},t\ge 0$, and related quantities such as the excess over the boundary $R_{t}=S_{\tau(t)}-t$. In particular, we determine the asymptotic distribution of $R_{t}$, as $t\to\i$, provide an asymptotic expansion of $E\tau(t)$ up to vanishing terms and are able to confirm a conjecture by Janson who earlier obtained related results.

Keywords: random walks,$m$-dependence,$(m+1)$-block factors,laddervariables,first passage times,excess over the boundary,Markov renewal theory,marked point process