Sur lʹapproximation de la distribution stationnaire dʹune chaîne de Markov stochastiquement monotone

Let P be an infinite irreducible stochastic matrix, stochastically dominated by an irreducible, positive-recurrent and stochastically monotone stochastic matrix Q. Let Pn be any n × n stochastic matrix with Pn ⩾ Tn, where Tn denotes the n × n northwest corner truncation of P. We first show that these assumptions imply the existence of limiting distributions μ, π, πn for Q, P, Pn respectively; moreover, if Q obeys a Foster-Lyapounov condition, we derive the rate of convergence of πn to π; as an application of the preceding results, we deal with the random walk on a half line, and prove under mild assumptions that the rate of convergence of πn to π is geometric.