Conditioning properties of the stationary distribution for a Markov chain

Abstract

Let T be an irreducible stochastic matrix with stationary vector π^T . The conditioning of π^T under perturbation of T is discussed by providing an attainable upper bound on the absolute value of the derivative of each entry in π^T with respect to a given perturbation matrix. Connections are made with an existing condition number for π^T , and the results are applied to the class of Markov chains arising from a random walk on a tree.