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Quasi-Stationary Distributions for Single Death Processes with Killing

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  • Zhe-Kang Fang

    (School of Mathematics and Statistics, Fujian Normal University
    Ministry of Education)

  • Yong-Hua Mao

    (School of Mathematics and Statistics, Fujian Normal University
    Ministry of Education)

Abstract

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.

Suggested Citation

  • Zhe-Kang Fang & Yong-Hua Mao, 2025. "Quasi-Stationary Distributions for Single Death Processes with Killing," Journal of Theoretical Probability, Springer, vol. 38(3), pages 1-51, September.
    • Handle: RePEc:spr:jotpro:v:38:y:2025:i:3:d:10.1007_s10959-025-01429-6
    DOI: 10.1007/s10959-025-01429-6
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    References listed on IDEAS

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    1. Foucart, Clément & Li, Pei-Sen & Zhou, Xiaowen, 2020. "On the entrance at infinity of Feller processes with no negative jumps," Statistics & Probability Letters, Elsevier, vol. 165(C).
    2. Velleret, Aurélien, 2022. "Unique quasi-stationary distribution, with a possibly stabilizing extinction," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 98-138.
    3. Fitzsimmons, P. J. & Pitman, Jim, 1999. "Kac's moment formula and the Feynman-Kac formula for additive functionals of a Markov process," Stochastic Processes and their Applications, Elsevier, vol. 79(1), pages 117-134, January.
    4. Yamato, Kosuke, 2025. "Existence of quasi-stationary distributions for downward skip-free Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 183(C).
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