Covariance and relaxation time in finite Markov chains. (1998)

• Record Type: Journal Article
• Authors: Keilson, Julian
• Is Part Of: Journal of applied mathematics and stochastic analysis. Volume 11:Number 3(1998)
• Page Start: 391
• Page End: 396
• Publication Date: 1998
• Subjects: finite Markov chains -- covariance -- relaxation time; Mathematical models -- Periodicals; Computer simulation -- Periodicals; Computer science -- Mathematics -- Periodicals; Computer science -- Mathematics; Computer simulation; Mathematical models; Applied Mathematics; Periodicals; Electronic journals; 519.22
• DOI: 10.1155/S104895339800032X
• Languages: English
• ISSNs: 1048-9533

Abstract

Abstract : The relaxation time T R E L 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 λ j of the infinitesimal generator Q ¯ ¯ . One uses T R E L = θ − 1 where θ = min λ j ≠ 0 { Re λ j [ − 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 ρ f ( τ ) has the bound | ρ f ( τ ) | ≤ exp [ − π | τ | ] and that this inequality is tight. The argument is almost entirely in the real domain.