Markov Renewal Theory for Stationary ( m + 1)-Block Factors: First Passage Time and Overshoot

Abstract

Given a sequence of i.i.d. random variables Y −m , Y −m+1,… with common distribution F and values in an arbitrary measurable space (S, 𝔖), let (S n ) n≥0 be a random walk whose increments X n are nonnegative (m + 1)-block factors of the form ϕ(Y n−m ,…, Y n ) for a measurable function ϕ:S m+1 → [0, ∞). 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 renewal process with stationary drift μ = E ϕ(Y −m ,…, Y 0). Suppose μ > 0, and let τ(t) = inf {n:S n > t } be the first passage time of (S n ) n≥0 beyond level t ≥ 0. An important variable related to τ(t) is the (m + 1)-step overshoot R m, t = S τ (t)+m − t which reduces to the familiar overshoot R t = S τ (t) − t if m = 0. The main results of this article are a second order approximation of the variance of τ(t) and bounds for for p ≥ 1 similar to those derived by Lorden in the i.i.d case (m = 0).