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 row-vector (interpreted as a probability-distribution).
Örjan Stenflo. "Perfect sampling from the limit of deterministic products of stochastic matrices." Electron. Commun. Probab. 13 474 - 481, 2008. https://doi.org/10.1214/ECP.v13-1409