Sensitivity of the Stationary Distribution of a Markov Chain

It is well known that if the transition matrix of an irreducible Markov chain of moderate size has a subdominant eigenvalue which is close to 1, then the chain is ill conditioned in the sense that there are stationary probabilities which are sensitive to perturbations in the transition probabilities. However, the converse of this statement has heretofore been unresolved. The purpose of this article is to address this issue by establishing upper and lower bounds on the condition number of the chain such that the bounding terms are functions of the eigenvalues of the transition matrix. Furthermore, it is demonstrated how to obtain estimates for the condition number of an irreducible chain with little or no extra computational effort over that required to compute the stationary probabilities by means of an LU or QR factorization.