On the perturbation of Markov chains with nearly transient states (Q1326399)

The author considers an irreducible stochastic matrix \(A\) where the `states' are thought of as composed of two consecutive sets, and the stationary distribution vector \({\underset \sim \pi}^ T = ({\underset \sim \pi}^ T_ 1,{\underset \sim \pi}^ T_ 2)\) , is partitioned accordingly. The situation of interest is where the states of the second set are almost transient. The author shows that only small relative perturbations in \({\underset \sim \pi}^ T_ 2\) are induced by perturbation of entries of \(A\) other than in \(E_{12}\), the one-step transition probabilities from the first to the second set.