Covariance and relaxation time in finite Markov chains

[摘要] The relaxation timeTRELof 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 eigenvaluesλjof the infinitesimal generatorQ¯¯. One usesTREL=θ−1whereθ=minλj≠0{Reλj[−Q¯¯]}. This paper establishes for the relaxation timeθ−1the theoretical solidity of the time reversible case. It does so by examining the structure of the quadratic distanced(t)to ergodicity. It is shown that, for any functionf(j)defined for statesj, the correlation functionρf(τ)hasthe bound|ρf(τ)|≤exp[−π|τ|]and that this inequality is tight. Theargument is almost entirely in the real domain.

[学科分类] 应用数学