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On the existence of moments of stopped sums in Markov renewal theory

Alsmeyer G.

Probability and Mathematical Statistics

2003

Vol. 23, Fasc. 2

389--411

Let (Mn)n ≥ 0 be an ergodic Markov chain on a general state space X with stationary distribution π and g: X → [0, ∞) a measurable function. Define S0 (g)def = 0 and Sn (g)def = g (M1) +…+ g (Mn) for n ≥ 1. Given any stopping time T for (Mn)n ≥ 0 and any initial distribution ν for (Mn)n ≥ 0, the purpose of this paper is to provide suitable conditions for the finiteness of Eν ST (g)p for p > 1. A typical result states that Eν ST (g)p ≤ C1 (Eν ST (gp) + Eν Tp) + C2 for suitable finite constants C1, C2. Our analysis is based to a large extent on martingale decompositions for Sn (g) and on drift conditions for the function g and the transition kernel P of the chain. Some of the results are stated under the stronger assumption that (Mn)n ≥ 0 is positive Harris recurrent in which case stopping times T which are regeneration epochs for the chain are of particular interest. The important special case where T = T(t)def = inf {n ≥ 1: Sn (g) > t} for t ≥ 0 is also treated.

The ladder variables of a Markov random walk

Alsmeyer G.

Probability and Mathematical Statistics

2000

Vol. 20, Fasc. 1

151--168

Given a Harris chain (Mn)n≥0 on any state space (S, C) with essentially unique stationary measure ξ, let (Xn)n≥0 be a sequence of real-valued random variables which are conditionally independent, given (Mn)n≥0, and satisfy [formula].. for some stochastic kernel Q : S2 × B → [0, 1] and all k ≥ 1. Denote by Sn the n-th partial sum of this sequence. Then (Mn, Sn)n≥0 forms a so-called Markov random walk with driving chain (Mn, Sn)n≥0. Its stationary mean drift is given by μ = EξX1 and assumed to be positive in which case the associated (strictly ascending) ladder epochs [formula].. and the ladder heights S*n = Sσn for n ≥ 0 are a.s. positive and finite randomvariables. Put M*n = Mσn. ……..[formula]