Probability and Mathematical Statistics: Vol. 23, Fasc. 2

ON THE EXISTENCE OF MOMENTS OF STOPPED SUMS IN MARKOV RENEWAL THEORY

Gerold Alsmeyer

Abstract: Let $(M ) n n>0$ be an ergodic Markov chain on a general state space $X$ with stationary distribution $p$ and $g: X --> [0, oo )$ a measurable function. Define $S (g) = 0 0$ and $def S(g) = g(M1) + ...+ g(Mn)$ for $n > 1.$ Given any stopping time $T$ for $(Mn)n>0$ and any initial distribution $n$ for $(Mn)n>0,$ the purpose of this paper is to provide suitable conditions for the finiteness of $p EnST (g)$ for $p > 1.$ A typical result states that

p p p EnST(g) < C1(EnST (g )+ EnT )+ 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 $T = T(t) d=ef inf(n > 1: S (g) > t) n$ for $t > 0$ is also treated.

2000 AMS Mathematics Subject Classification: 60K1S, 60G42, 60G40.

Key words and phrases: Markov random walk, stopped sum, Harris recurrence, regeneration epoch, drift condition, $l$ -dependence, martingale, Burkholder’s inequality.