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 with P-n greater than or equal to T-n, where T-n denotes the n x n northwest corner truncation of P. These assumptions imply the existence of limit distributions pi and pi(n) for P and P-n respectively. We show that 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 convergence of the generating function of pi. As an application of the preceding 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 that if the i.i.d. input sequence (A(m)) is such that we can find a real 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.