Journal of Applied Mathematics and Stochastic Analysis
Volume 11 (1998), Issue 3, Pages 283-288

Monotone measures of ergodicity for Markov chains

J. Keilson1 and O. A. Vasicek1

1University of Rochester, William E. Simon Graduate School of Business Administration, Rochester, NY 14627, USA
2KMV Corporation, San Francisco, CA, USA

Received 1 October 1997; Revised 1 February 1998

Copyright © 1998 J. Keilson and O. A. Vasicek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The following paper, first written in 1974, was never published other than as part of an internal research series. Its lack of publication is unrelated to the merits of the paper and the paper is of current importance by virtue of its relation to the relaxation time. A systematic discussion is provided of the approach of a finite Markov chain to ergodicity by proving the monotonicity of an important set of norms, each measures of egodicity, whether or not time reversibility is present. The paper is of particular interest because the discussion of the relaxation time of a finite Markov chain [2] has only been clean for time reversible chains, a small subset of the chains of interest. This restriction is not present here. Indeed, a new relaxation time quoted quantifies the relaxation time for all finite ergodic chains (cf. the discussion of Q1(t) below Equation (1.7)]. This relaxation time was developed by Keilson with A. Roy in his thesis [6], yet to be published.