Sensitivity of the stationary distribution vector for an ergodic Markov chain

Abstract

Stationary distribution vectors p ∞ for Markov chains with associated transition matrices T are important in the analysis of many models in the mathematical sciences, such as queuing networks, input-output economic models, and compartmental tracer analysis models. The purpose of this paper is to provide insight into the sensitivity of p ∞ to perturbations in the transition probabilities of T and to understand some of the difficulties in computing an accurate p ∞. The group inverse A # of I − T is shown to be of fundamental importance in understanding sensitivity or conditioning of p ∞. 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 ∞ cannot be sensitive to small perturbations in T. Ecological examples are given. A new algorithm for calculating A # is described.