P.K. Pollett

Abstract: In a recent paper, Lenin et al.^* introduced the idea of similarity in the context of birth-death processes. The present note examines the extent to which their results can be extended to arbitrary Markov chains. We prove, under a variety of conditions, that similar chains are weakly similar in a sense which we shall describe, and we prove that minimal chains are weakly similar if and only if the corresponding transition-rate matrices are weakly similar. We provide a general framework for constructing families of weakly similar chains, one which allows us to construct all such chains in the irreducible case.

^* R.B. Lenin, P.R. Parthasarathy, W.R.W. Scheinhardt, and E.A. van Doorn. (2000) Families of birth-death processes with similar time-dependent behavour. J. Appl. Probab. 37, 835--849.

Keywords: Invariant measures; birth-death processes

Acknowledgement: This worked was funded by the Australian Research Council.

The author:

• Phil Pollett, Discipline of Mathematics, The University of Queensland. pkp@maths.uq.edu.au

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Last modified: 26th January 2000