Augmented truncation approximations of discrete-time Markov chains

Let PP be a positive recurrent infinite transition matrix with invariant distribution ππ and (n)P̃ be a truncated and arbitrarily augmented stochastic matrix with invariant distribution (n)π(n)π. We investigate the convergence ‖(n)π−π‖→0‖(n)π−π‖→0, as n→∞n→∞, and derive a widely applicable sufficient criterion. Moreover, computable bounds on the error ‖(n)π−π‖‖(n)π−π‖ are obtained for polynomially and geometrically ergodic chains. The bounds become rather explicit when the chains are stochastically monotone.