Similar Markov chains
Phil Pollett
Abstract:
In a recent paper (J. Appl. Probab. 37 (2000), 835849) Lenin,
Parthasarathy, Scheinhardt and van Doorn introduced the idea of
similarity in the context of birthdeath processes. I will examine the
extent to which their results can be extended to arbitrary
continuoustime Markov chains over a countable state space S. Two such
chains are said to be similar
if their transition functions P and
satisfy
for some collection of positive constants c_{ij},
(p_{ij}(t) can be interpretted as the probability of moving from state
i to state j in time t). I will prove, under a variety of
conditions, that similar chains are strongly similar in the
following sense:
for positive constants
.
I will
also show that minimal chains are strongly similar if and only if the
corresponding transitionrate matrices are also strongly similar in an
obvious sense. This gives rise to a general framework for constructing
families of strongly similar chains, one which allows us to construct
all such chains in the irreducible case. I will draw heavily on the
"VereJones theory" of Markov chains, which provides the basic
framework for studying questions concerning the convergence of Markov
transition probabilities and their associated spectral properties.
Acknowledgement:
This worked was funded by
the Australian Research Council.
The author:

Phil Pollett,
Department of Mathematics,
The University of Queensland.
pkp@maths.uq.edu.au
Last modified: 17th April 2001