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Comparison theorems for reversible Markov chains. (English) Zbl 0799.60058

This paper develops methods for getting upper and lower bounds on the eigenvalues \[ 1=\beta_ 0>\beta_ 1\geq\cdots\geq\beta_{| X|- 1}\geq-1, \] of an irreducible reversible Markov matrix \(P\) by comparison with a second reversible chain on the same state space \(X\). It extends the ideas introduced by the authors [Ann. Probab. 21, No. 4, 2131-2156 (1993; Zbl 0790.60011)], where the random walks on finite groups were considered, and the geometric properties as the diameter and covering number of an associated graph, which appeared in [first author and D. Stroock, Ann. Appl. Probab. 1, No. 1, 36-61 (1991; Zbl 0731.60061)] are utilized. Apparently the bounds given in the paper have wide applications, especially in exclusion processes.
Reviewer: W.-Z.Yang (Taipei)

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60F05 Central limit and other weak theorems
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