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Inverse Problems for Ergodicity of Markov Chains

Zhi-Feng Wei

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.

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 -matrix (or transition matrix 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.

Paper Structure

This paper contains 10 sections, 30 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Proofs of Criteria for Inverse Problems
  3. Minimal Solution Theory Preparations
  4. Lower Bound for Polynomial Moments and Sufficiency
  5. Approximation for Polynomial Moments and Necessity
  6. Proof of \ref{['in_eerg_con']}
  7. Applications to Single Birth Processes
  8. Explicit Criteria for Single Birth Processes: Alternative Proofs
  9. A Special Class of Single Birth Processes
  10. Applications to multi-dimensional examples

Key Result

Theorem 1

Let $Q$ be an irreducible regular $Q$-matrix and $H$ a non-empty finite subset of a countable state space $E$. Then the $Q$-process is strongly ergodic if and only if there exists a bounded solution $(y_i)_{i\in E}$ to inequality

Theorems & Definitions (63)

  • Theorem : hou1988twd1981
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Example
  • Example
  • Theorem \ref{in_erg_con}$^\prime$
  • Theorem \ref{in_serg_con}$^\prime$
  • ...and 53 more