Convergence Property of Standard Transition Functions

A standard transition function P = (P ij (t)) is called ergodic (positive recurrent) if there exists a probability measure π = (π i ; i ∈ E) such that $$ \mathop{{\lim }}\limits_{{t \to \infty }} {{p}_{i}}_{j}(t) = {{\pi }_{j}} > 0,\forall i \in E $$ (0.1) The aim of this paper is to discuss the convergence problem in (0.1). We shall study four special types of convergence: the so-called strong ergodicity, uniform polynomial convergence, L 2-exponential ergodicity and exponential ergodicity. Our main interest is always to characterize these properties in terms of the q-matrix.