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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.