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<p>The paper focuses on the ergodicity of a <i>ϕ</i>-irreducible Markov chain {<i>X</i><sub><i>n</i></sub>, <i>n</i> ≥ 0} that is generated iteratively through the expression <i>X</i><sub><i>n</i>+1</sub> = <i>f</i>(<i>X</i><sub><i>n</i></sub>) + <i>ϵ</i><sub><i>n</i>+1</sub>. Here, {<i>ϵ</i><sub><i>n</i></sub>, <i>n</i> ≥ 1} is a sequence of independent identically distributed centered random variables, <i>f</i>(·) is an ℝ-valued continuous function, and <i>X</i><sub>0</sub> is arbitrary but independent of {<i>ϵ</i><sub><i>n</i></sub>, <i>n</i> ≥ 1}. Our main contribution is to provide necessary and sufficient conditions for the ergodicity of this special class of Markov chains. We also present a generalized approach for <i>f</i>(·) in the end.</p>

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Ergodicity of a Special Class of Markov Chains

  • 期刊论文

  • 发表日期：

    2026年1月2日

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  • Xin-yu Hu,
  • Ping He

摘要

The paper focuses on the ergodicity of a ϕ-irreducible Markov chain {Xn, n ≥ 0} that is generated iteratively through the expression Xn+1 = f(Xn) + ϵn+1. Here, {ϵn, n ≥ 1} is a sequence of independent identically distributed centered random variables, f(·) is an ℝ-valued continuous function, and X0 is arbitrary but independent of {ϵn, n ≥ 1}. Our main contribution is to provide necessary and sufficient conditions for the ergodicity of this special class of Markov chains. We also present a generalized approach for f(·) in the end.