Inverse Problems for Ergodicity of Markov Chains

TL;DR

This paper develops criteria for solving inverse ergodicity problems in Markov Chains, using inequalities involving the transition or Q-matrix, applicable to various models including birth processes.

Contribution

It introduces new criteria based on inequalities for the Q-matrix to address inverse ergodicity problems in both continuous and discrete-time Markov Chains.

Findings

Criteria established for various ergodicity types

Applicable to single birth and multi-dimensional models

Provides universal treatment for inverse ergodicity problems

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.