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On Separation for Birth-Death Processes

This article considers separation for a birth-death process on a finite state space S = [1,2,…, N]. Separation is defined by si(t) = 1 – minj∈sPij(t)/πj, as in Fill [5,6], where Pij(t) denotes the transition probabilities of the birth-death process and πj the stationary probabilities. Separation is a measure of nonstationarity of Markov chains and provides an upper bound of the variation distance. Easily computable upper bounds for si-(t) are given, which consist of simple exponential functions whose parameters are the eigenvalues of the infinitesimal generator or its submatrices of the birth-death process.