Perfect sampling from the limit of deterministic products of stochastic matrices

We illustrate how a technique from the theory of random iterations of functions can be used within the theory of products of matrices. Using this technique we give a simple proof of a basic theorem about the asymptotic behavior of (deterministic) “backwards products” of row-stochastic matrices and present an algorithm for perfect sampling from the limiting common rowvector (interpreted as a probability-distribution). 1. Preliminaries An N ×N matrix P = (pij) is called stochastic if ∑ j pij = 1, and pij ≥ 0 for all i and j. A stochastic matrix P is called stochastic-indecomposable-aperiodic (SIA) if Q = lim n→∞ P exists and has all rows equal. Such matrices are of interest in the theory of Markov chains since if {Xn} is a Markov chain with a transition matrix, P , which is SIA then in the long run Xn will be distributed according to the common row vector of Q independent of the value of X0. Basic theory of Markov chains gives soft conditions ensuring a stochastic matrix to be SIA. Essentially “periodicity” has to be ruled out, and one fixed closed irreducible set should eventually be reached by the chain from any given starting point. To have something in mind, the matrix