Inverse Problems for Ergodicity of Markov Chains

Abstract

For both continuous-time and discrete-time Markov Chains, we provide criteria for inverse problems of classical types of ergodicity: (ordinary) erogodicity, algebraic ergodicity, exponential ergodicity and strong ergodicity. Our criteria are in terms of the existence of solutions to inequalities involving the $Q$-matrix (or transition matrix $P$ in time-discrete case) of the process. Meanwhile, these criteria are applied to some examples and provide "universal" treatment, including single birth processes and several multi-dimensional models.