Sensitivity of the stationary distribution vector for an ergodic Markov chain (Q1069213)

Stationary distribution vectors \(p^{\infty}\) for Markov chains with associated transition matrices T are important in the analysis of many models in the mathematical sciences, such as queueing networks, input- output economic models, and compartmental tracer analysis models. The purpose of this paper is to provide insight into the sensitivity of \(p^{\infty}\) to perturbations in the transition probabilities of T and to understand some of the difficulties in computing an accurate \(p^{\infty}\). The group inverse \(A^{\#}\) of I-T is shown to be of fundamental importance in understanding sensitivity or conditioning of \(p^{\infty}.\) The main result shows that if there is a state that is accessible from every other state and the corresponding column of T has no small off- diagonal elements, then \(p^{\infty}\) cannot be sensitive to small perturbations in T. Ecological examples are given. A new algorithm for calculating \(A^{\#}\) is described.