Journal of Applied Mathematics and Stochastic Analysis
Volume 11 (1998), Issue 3, Pages 391-396
doi:10.1155/S104895339800032X
Covariance and relaxation time in finite Markov chains
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 Julian Keilson. 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.