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Non-(strong, geometrically) ergodicity criteria for discrete time Markov chains on general state

Ling-Di Wang, Yu Chen, Yu-Hui Zhang

TL;DR

This work develops inverse-problem criteria for non-ergodicity, non-strong ergodicity, and non-geometric ergodicity of discrete-time Markov chains on general state spaces by exploiting minimal nonnegative solutions to drift-type inequalities involving the one-step kernel. The authors establish a rigorous framework based on minimal solution theory, Dynkin's formula, and Lyapunov-type conditions, and they introduce an approximation scheme using increasing compact sets to derive necessity results and address geometric ergodicity. The methods are then applied to a broad immigration-augmented class of Markov processes, yielding explicit, verifiable criteria for recurrence, ergodicity, and their non-geometric/non-strong variants. The paper also provides two concrete examples—a general immigration birth process and a birth–death–immigration model—illustrating how the criteria translate into computable conditions. Overall, the results offer practical tools for diagnosing non-stability in complex Markov models relevant to applied probability and stochastic processes.

Abstract

For discrete-time Markov chains on general state spaces, we establish criteria for non-ergodicity and non-strong ergodicity, and derive sufficient conditions for non-geometric ergodicity via the theory of minimal nonnegative solutions. Our criteria are formulated based on the existence of solutions to inequalities involving the chain's one-step transition kernel. Meanwhile, these practical criteria are applied to a type of examples, which can effectively characterize the non-ergodicity and non-strong ergodicity of a specific class of single birth (death) processes.

Non-(strong, geometrically) ergodicity criteria for discrete time Markov chains on general state

TL;DR

This work develops inverse-problem criteria for non-ergodicity, non-strong ergodicity, and non-geometric ergodicity of discrete-time Markov chains on general state spaces by exploiting minimal nonnegative solutions to drift-type inequalities involving the one-step kernel. The authors establish a rigorous framework based on minimal solution theory, Dynkin's formula, and Lyapunov-type conditions, and they introduce an approximation scheme using increasing compact sets to derive necessity results and address geometric ergodicity. The methods are then applied to a broad immigration-augmented class of Markov processes, yielding explicit, verifiable criteria for recurrence, ergodicity, and their non-geometric/non-strong variants. The paper also provides two concrete examples—a general immigration birth process and a birth–death–immigration model—illustrating how the criteria translate into computable conditions. Overall, the results offer practical tools for diagnosing non-stability in complex Markov models relevant to applied probability and stochastic processes.

Abstract

For discrete-time Markov chains on general state spaces, we establish criteria for non-ergodicity and non-strong ergodicity, and derive sufficient conditions for non-geometric ergodicity via the theory of minimal nonnegative solutions. Our criteria are formulated based on the existence of solutions to inequalities involving the chain's one-step transition kernel. Meanwhile, these practical criteria are applied to a type of examples, which can effectively characterize the non-ergodicity and non-strong ergodicity of a specific class of single birth (death) processes.

Paper Structure

This paper contains 11 sections, 23 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Proof of criteria for inverse problems
  3. Recall minimal solution theory
  4. Lower bounds for moments of hitting times and sufficiency
  5. Approximation for the moments of hitting times; necessity of Theorems \ref{['Non-1-ergo']} \ref{['Non-stro']} and proof of Theorem \ref{['Non-geo']}
  6. Approximation of Markov chain
  7. Necessity of Theorems \ref{['Non-1-ergo']} and \ref{['Non-stro']}
  8. Proof of Theorem \ref{['Non-geo']}
  9. A class of Markov process with immigration
  10. Example 1
  11. Example 2

Key Result

Theorem 1.1

(Non-ergodicity) Assume that the Markov chain $\{X_n\}$ satisfies Assumption 1. Then $\{X_n\}$ is not ergodic if there exist a set $A \in \mathscr{B}^+(\mathscr{X})$ and a sequence of functions $\{V^{(n)}(x)\}_{n \ge 1}$ with $V^{(n)}(x): \mathscr{X} \to \mathbb{R}$ that satisfy the following condit The converse also holds when the Markov chain $\{X_n\}$ is recurrent.

Theorems & Definitions (29)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Proposition 1.8
  • Remark 1.9
  • Definition 2.1
  • ...and 19 more