Stationary distributions and mean first passage times of perturbed Markov chains

For finite irreducible discrete time Markov chains, whose transition probabilities are subjected to a perturbation, it is shown that the mean first passage times play an important role in determining the differences between the stationary probabilities in the perturbed and unperturbed situations. The derivation of the interconnection, under the updating procedure, is explored through the use of generalized matrix inverses. New improved bounds for the relative and absolute differences between the stationary probabilities are derived. Some interesting qualitative results regarding the nature of the relative and absolute changes to the stationary probabilities are also obtained. Similar procedures are used to establish an updating procedure for mean first passage times under perturbations.