Covariance and relaxation time in finite Markov chains

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.