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Journal of Applied Mathematics and Stochastic Analysis
Volume 11, Issue 3, Pages 391-396
http://dx.doi.org/10.1155/S104895339800032X

Covariance and relaxation time in finite Markov chains

Julian Keilson

University of Rochester, William E. Simon Graduate School of Business Administration, Rochester 14627, NY, USA

Received 1 February 1998; Revised 1 March 1998

Copyright © 1998 Hindawi Publishing Corporation. 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.

Abstract

The relaxation time TREL of a finite ergodic Markov chain in continuous time, i.e., the time to reach ergodicity from some initial state distribution, is loosely given in the literature in terms of the eigenvalues λj of the infinitesimal generator Q¯¯. One uses TREL=θ−1 where θ=minλj≠0{Reλj[−Q¯¯]}. This paper establishes for the relaxation time θ−1 the theoretical solidity of the time reversible case. It does so by examining the structure of the quadratic distance d(t) to ergodicity. It is shown that, for any function f(j) defined for states j, the correlation function ρf(τ) has the bound |ρf(τ)|≤exp[−π|τ|] and that this inequality is tight. The argument is almost entirely in the real domain.