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The Arsenal of Perturbation Bounds for Finite Continuous-Time Markov Chains: A Perspective

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  • Alexander Y. Mitrophanov

    (Frederick National Laboratory for Cancer Research, National Institutes of Health, Frederick, MD 21702, USA)

Abstract

Perturbation bounds are powerful tools for investigating the phenomenon of insensitivity to perturbations, also referred to as stability, for stochastic and deterministic systems. This perspective article presents a focused account of some of the main concepts and results in inequality-based perturbation theory for finite state-space, time-homogeneous, continuous-time Markov chains. The diversity of perturbation bounds and the logical relationships between them highlight the essential stability properties and factors for this class of stochastic processes. We discuss the linear time dependence of general perturbation bounds for Markov chains, as well as time-independent (i.e., time-uniform) perturbation bounds for chains whose stationary distribution is unique. Moreover, we prove some new results characterizing the absolute and relative tightness of time-uniform perturbation bounds. Specifically, we show that, in some of them, an equality is achieved. Furthermore, we analytically compare two types of time-uniform bounds known from the literature. Possibilities for generalizing Markov-chain stability results, as well as connections with stability analysis for other systems and processes, are also discussed.

Suggested Citation

  • Alexander Y. Mitrophanov, 2024. "The Arsenal of Perturbation Bounds for Finite Continuous-Time Markov Chains: A Perspective," Mathematics, MDPI, vol. 12(11), pages 1-15, May.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:11:p:1608-:d:1398433
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    References listed on IDEAS

    as
    1. Zeifman, A. I. & Isaacson, Dean L., 1994. "On strong ergodicity for nonhomogeneous continuous-time Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 50(2), pages 263-273, April.
    2. Bouranis, Lampros & Friel, Nial & Maire, Florian, 2018. "Model comparison for Gibbs random fields using noisy reversible jump Markov chain Monte Carlo," Computational Statistics & Data Analysis, Elsevier, vol. 128(C), pages 221-241.
    3. Guglielmo D’Amico & Riccardo De Blasis & Fulvio Gismondi, 2023. "Perturbation analysis for dynamic poverty indexes," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 52(19), pages 6820-6839, October.
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