THE LADDER VARIABLES OF A MARKOV RANDOM WALK
Abstract: Given a Harris chain on any state space with essentially unique
stationary measure let be a sequence of real-valued random variables which
are conditionally independent, given and satisfy
for
some stochastic kernel
and all
Denote by
the
-th partial
sum of this sequence. Then
forms a so-called Markov random walk with
driving chain
Its stationary mean drift is given by
and
assumed to be positive in which case the associated (strictly ascending) ladder
epochs
and the ladder heights
for
are a.s. positive and finite random
variables. Put
The main result of this paper is that
and
are again Markov random walks (with positive increments, thus so-called
Markov renewal processes) with Harris recurrent driving chain
The
difficult part is to verify the Harris recurrence of
Denoting by
its
stationary measure, we also give necessary and sufficient conditions for the finiteness of
and
in terms of
or the recurrence-type of
or
1991 AMS Mathematics Subject Classification: 60J05, 60J15, 60K05, 60K15.
Key words and phrases: Markov random walks, ladder variables, Harris recurrence,
regeneration epochs, couphng.