This paper discusses perturbation bounds for the stationary distribution of a finite indecomposable Markov chain. Existing bounds are reviewed. New bounds are presented which more completely exploit the stochastic features of the perturbation and which also are easily computable. Examples illustrate the tightness of the bounds and their application to bounding the error in the Simon–Ando aggregation technique for approximating the stationary distribution of a nearly completely decomposable Markov chain.

Let P denote an irreducible positive recurrent infinite stochastic matrix with the unique invariant probability measure π. We consider sequences {Pm}m∊N of stochastic matrices converging to P (pointwise), such that every Pm has at least one invariant probability measure πm. The aim of this paper is to find conditions, which assure that at least one of sequences {πm}m∊N converges to π (pointwise). This includes the case where the Pm are finite matrices, which is of special interest. It is shown that there is a sequence of finite stochastic matrices, which can easily be constructed, such that {πm}m∊N converges to π. The conditions given for the general case are closely related to Foster's condition.

It is shown that a class of infinite, block-partitioned, stochastic matrices has a matrix-geometric invariant probability vector of the form (x0, x1,…), where xk = x0Rk, for k ≧ 0. The rate matrix R is an irreducible, non-negative matrix of spectral radius less than one. The matrix R is the minimal solution, in the set of non-negative matrices of spectral radius at most one, of a non-linear matrix equation.

Applications to queueing theory are discussed. Detailed explicit and computationally tractable solutions for the GI/PH/1 and the SM/M/1 queue are obtained.

The problem of tendency to consensus in an information-exchanging operation is connected with the ergodicity problem for backwards products of stochastic matrices. For such products, weak and strong ergodicity, defined analogously to these concepts for forward products of inhomogeneous Markov chain theory, are shown (in contrast to that theory) to be equivalent. Conditions for ergodicity are derived and their relation to the consensus problem is considered.

The paper is concerned with aspects of the behaviour of the products of finite stochastic matrices, the methods used in the proofs being of a probabilistic nature. The main result of the paper (Theorem 1) presents a general picture of the asymptotic behaviour of the transition probabilities between various groups of states. A unified treatment of some results of non-homogeneous Markov chain theory pertaining to weak ergodicity is then given.