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<p>This paper studies the quasi-stationary distributions for a single death process (or downwardly skip-free process) with killing defined on the nonnegative integers, corresponding to a non-conservative transition rate matrix. The set <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10959_2025_1429_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="89" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{1,2,3,\ldots \}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> constitutes an irreducible class, and 0 is an absorbing state. For the single death process with three kinds of killing term, we obtain the existence and uniqueness of the quasi-stationary distribution. Moreover, we derive the conditions for exponential convergence to the quasi-stationary distribution in the total variation norm. Our main approach is based on the Doob’s <i>h</i>-transform, potential theory and probabilistic methods.</p>

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Quasi-Stationary Distributions for Single Death Processes with Killing

  • 期刊论文

  • 发表日期：

    2025年7月10日

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  • Zhe-Kang Fang,
  • Yong-Hua Mao

摘要

This paper studies the quasi-stationary distributions for a single death process (or downwardly skip-free process) with killing defined on the nonnegative integers, corresponding to a non-conservative transition rate matrix. The set \(\{1,2,3,\ldots \}\) { 1 , 2 , 3 , … } constitutes an irreducible class, and 0 is an absorbing state. For the single death process with three kinds of killing term, we obtain the existence and uniqueness of the quasi-stationary distribution. Moreover, we derive the conditions for exponential convergence to the quasi-stationary distribution in the total variation norm. Our main approach is based on the Doob’s h-transform, potential theory and probabilistic methods.