Probability and Mathematical Statistics: Vol. 20, Fasc. 1

THE LADDER VARIABLES OF A MARKOV RANDOM WALK

Gerold Alsmeyer

Abstract: Given a Harris chain $(M ) nn>0$ on any state space $(S,C)$ with essentially unique stationary measure $q,$ let $(X ) nn>0$ be a sequence of real-valued random variables which are conditionally independent, given $(M ) , n n>0$ and satisfy

P(X (- .|(M ) ) = Q(M ,M ,.) k nn>0 k-1 k

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 $(Mn)n>0.$ Its stationary mean drift is given by $m = EqX1$ and assumed to be positive in which case the associated (strictly ascending) ladder epochs

s0 = inf(k > 0 : Sk > 0),

sn = inf(k > sn- 1 : Sk > Ssn-1) forn > 1,

and the ladder heights $S*n = Ssn$ for $n > 0$ are a.s. positive and finite random variables. Put $M *n = Msn.$ The main result of this paper is that $(Mn*,S*n)n>0$ and $(M *n,sn)n>0$ are again Markov random walks (with positive increments, thus so-called Markov renewal processes) with Harris recurrent driving chain $(Mn*)n>0.$ The difficult part is to verify the Harris recurrence of $(M n*)n>0.$ Denoting by $q*$ its stationary measure, we also give necessary and sufficient conditions for the finiteness of $Eq*S*1,$ $EqS*1$ and $Eq*s1$ in terms of $m$ or the recurrence-type of $(Mn)n >0$ or $(M *n)n>0.$

1991 AMS Mathematics Subject Classification: 60J05, 60J15, 60K05, 60K15.

Key words and phrases: Markov random walks, ladder variables, Harris recurrence, regeneration epochs, couphng.