CHARACTERIZATION OF CONVERGENCE-RATES FOR THE APPROXIMATION OF THE STATIONARY DISTRIBUTION OF INFINITE MONOTONE STOCHASTIC MATRICES

Let P be an infinite irreducible stochastic matrix, recurrent positive and stochastically monotone and P-n be any n x n stochastic matrix wi th P-n greater than or equal to T-n, where T-n denotes the n x n north west corner truncation of P. These assumptions imply the existence of limit distributions pi and pi(n) for P and P-n respectively. We show t hat if the Markov chain with transition probability matrix P meets the further condition of geometric recurrence then the exact convergence rate of pi(n) to pi can be expressed in terms of the radius of converg ence of the generating function of pi. As an application of the preced ing result, we deal with the random walk on a half line and prove that the assumption of geometric recurrence can be relaxed. We also show t hat if the i.i.d. input sequence (A(m)) is such that we can find a rea l number r(0) > 1 with E{r(0)(A)) = 1, then the exact convergence rate of pi(n) to pi is characterized by lb. Moreover, when the generating function of A is not defined for \z\ > 1, we derive an upper bound for the distance between pi(n), and pi based on the moments of A.