Interlacing Eigenvalues in Time Reversible Markov Chains

Abstract
For irreducible, reversible, finite state chains we observe that two sets of eigenvalues related to the transition rate matrix Q, (λ0, …, λ m ) and (γ1, …, γ m ), are interlaced so that λ0 < γ1 < λ1 < ⋯ < γ m < λ m . Many quantities associated with ℒπ TA , the distribution of the first passage time to A starting in steady state, can be simply expressed in terms of these eigenvalues, and the interlacing property can be exploited to obtain approximations. There is a close connection between interlacing eigenvalues and completely monotone distributions, as well as a representation for finite mixtures of exponential distributions, which follow from this observation.

This publication has 12 references indexed in Scilit: