Similar Markov chains

Phil Pollett

Abstract: In a recent paper (J. Appl. Probab. 37 (2000), 835-849) Lenin, Parthasarathy, Scheinhardt and van Doorn introduced the idea of similarity in the context of birth-death processes. I will examine the extent to which their results can be extended to arbitrary continuous-time Markov chains over a countable state space S. Two such chains are said to be similar if their transition functions P and $\widetilde P$ satisfy

\begin{displaymath}\tilde p_{ij}(t)=c_{ij} p_{ij}(t), \qquad i,j\in S, \ t>0,

for some collection of positive constants cij, $i,j\in S$ (pij(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:

\begin{displaymath}\tilde p_{ij}(t)=\alpha_i \beta_j p_{ij}(t), \qquad i,j\in S, \ t>0,

for positive constants $\alpha_i, \beta_i,i\in C$. I will also show that minimal chains are strongly similar if and only if the corresponding transition-rate 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 "Vere-Jones 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.

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Last modified: 17th April 2001