The paper deals mostly with continuous-time Markov chains and provides criteria for non-ergodicity, nonalgebraic ergodicity, non-exponential ergodicity, and non-strong ergodicity. For discrete-time Markov chains, it contains criteria for non-ergodicity, non-algebraic ergodicity, and non-strong ergodicity. The criteria are given in terms of the existence of solutions to inequalities involving the \(Q\)-matrix (or transition matrix \(P\) in time-discrete case) of the chain. Theorem 1 represents a typical result of the paper. Let \(\Pi = \left( {{\Pi _{ij}}:i,j \in E} \right)\) be the matrix of the embedded chain of the \(Q\)-process.
Theorem 1. Let \(Q\) be an irreducible regular \(Q\)-matrix and \(H\) be a non-empty finite subset of \(E\). Then the \(Q\)-process is non-ergodic iff there exists a sequence \(\left\{ {{y^{(n)}}} \right\}_{n = 1}^\infty \), where \({y^{(n)}} = {(y_i^{(n)})_{i \in E}}\) for each \(n \ge 1\), and \(\left\{ {{y^{(n)}}} \right\}_{n = 1}^\infty \) satisfies the following conditions: (1) for each \(n \ge 1\), \({y^{(n)}} = {(y_i^{(n)})_{i \in E}}\) satisfies \({\sup _{i \in E}}y_i^{(n)} < \infty \) and solves the inequality \({y_i} \le \sum\limits_{j \notin H} {{\Pi _{ij}}{y_j}} + \frac{1}{{{q_i}}}\),\(i \in E\); (2) \({\sup _{n \ge 1}}{\max _{i \in H}}y_i^{(n)} = \infty \).
The author applies these results to a special class of single birth processes and several multidimensional models.