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Approximating Markov chains and V-geometric ergodicity via weak perturbation theory

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  • Hervé, Loïc
  • Ledoux, James

Abstract

Let P be a Markov kernel on a measurable space X and let V:X→[1,+∞). This paper provides explicit connections between the V-geometric ergodicity of P and that of finite-rank non-negative sub-Markov kernels P̂k approximating P. A special attention is paid to obtain an efficient way to specify the convergence rate for P from that of P̂k and conversely. Furthermore, explicit bounds are obtained for the total variation distance between the P-invariant probability measure and the P̂k-invariant positive measure. The proofs are based on the Keller–Liverani perturbation theorem which requires an accurate control of the essential spectral radius of P on usual weighted supremum spaces. Such computable bounds are derived in terms of standard drift conditions. Our spectral procedure to estimate both the convergence rate and the invariant probability measure of P is applied to truncation of discrete Markov kernels on X:=N.

Suggested Citation

  • Hervé, Loïc & Ledoux, James, 2014. "Approximating Markov chains and V-geometric ergodicity via weak perturbation theory," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 613-638.
    • Handle: RePEc:eee:spapps:v:124:y:2014:i:1:p:613-638
    DOI: 10.1016/j.spa.2013.09.003
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    References listed on IDEAS

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    1. Rosenthal, Jeffrey S., 1996. "Markov chain convergence: From finite to infinite," Stochastic Processes and their Applications, Elsevier, vol. 62(1), pages 55-72, March.
    2. Roberts, G. O. & Tweedie, R. L., 1999. "Bounds on regeneration times and convergence rates for Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 80(2), pages 211-229, April.
    3. Robert B. Lund & Richard L. Tweedie, 1996. "Geometric Convergence Rates for Stochastically Ordered Markov Chains," Mathematics of Operations Research, INFORMS, vol. 21(1), pages 182-194, February.
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    Cited by:

    1. Badredine Issaadi, 2020. "Weak stability bounds for approximations of invariant measures with applications to queueing," Methodology and Computing in Applied Probability, Springer, vol. 22(1), pages 371-400, March.
    2. Hervé, Loïc & Ledoux, James, 2016. "A computable bound of the essential spectral radius of finite range Metropolis–Hastings kernels," Statistics & Probability Letters, Elsevier, vol. 117(C), pages 72-79.
    3. Loic Hervé & James Ledoux, 2020. "State-Discretization of V-Geometrically Ergodic Markov Chains and Convergence to the Stationary Distribution," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 905-925, September.

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