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Polynomial ergodicity of Markov transition kernels

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  • Fort, G.
  • Moulines, E.

Abstract

This paper discusses quantitative bounds on the convergence rates of Markov chains, under conditions implying polynomial convergence rates. This paper extends an earlier work by Roberts and Tweedie (Stochastic Process. Appl. 80(2) (1999) 211), which provides quantitative bounds for the total variation norm under conditions implying geometric ergodicity. Explicit bounds for the total variation norm are obtained by evaluating the moments of an appropriately defined coupling time, using a set of drift conditions, adapted from an earlier work by Tuominen and Tweedie (Adv. Appl. Probab. 26(3) (1994) 775). Applications of this result are then presented to study the convergence of random walk Hastings Metropolis algorithm for super-exponential target functions and of general state-space models. Explicit bounds for f-ergodicity are also given, for an appropriately defined control function f.

Suggested Citation

  • Fort, G. & Moulines, E., 2003. "Polynomial ergodicity of Markov transition kernels," Stochastic Processes and their Applications, Elsevier, vol. 103(1), pages 57-99, January.
    • Handle: RePEc:eee:spapps:v:103:y:2003:i:1:p:57-99
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    References listed on IDEAS

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    1. 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.
    2. Jarner, Søren Fiig & Hansen, Ernst, 2000. "Geometric ergodicity of Metropolis algorithms," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 341-361, February.
    3. Fort, Gersende & Moulines, Eric, 2000. "V-Subgeometric ergodicity for a Hastings-Metropolis algorithm," Statistics & Probability Letters, Elsevier, vol. 49(4), pages 401-410, October.
    4. O. Stramer & R. L. Tweedie, 1999. "Langevin-Type Models I: Diffusions with Given Stationary Distributions and their Discretizations," Methodology and Computing in Applied Probability, Springer, vol. 1(3), pages 283-306, October.
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    Citations

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    Cited by:

    1. Connor, S.B. & Fort, G., 2009. "State-dependent Foster-Lyapunov criteria for subgeometric convergence of Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 119(12), pages 4176-4193, December.
    2. Gareth O. Roberts & Jeffrey S. Rosenthal, 2011. "Quantitative Non-Geometric Convergence Bounds for Independence Samplers," Methodology and Computing in Applied Probability, Springer, vol. 13(2), pages 391-403, June.
    3. Meitz, Mika & Saikkonen, Pentti, 2022. "Subgeometrically Ergodic Autoregressions," Econometric Theory, Cambridge University Press, vol. 38(5), pages 959-985, October.
    4. Gareth O. Roberts & Jeffrey S. Rosenthal, 2023. "Polynomial Convergence Rates of Piecewise Deterministic Markov Processes," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-18, March.
    5. Mika Meitz & Pentti Saikkonen, 2022. "Subgeometrically ergodic autoregressions with autoregressive conditional heteroskedasticity," Papers 2205.11953, arXiv.org, revised Apr 2023.
    6. Mika Meitz & Pentti Saikkonen, 2019. "Subgeometric ergodicity and $\beta$-mixing," Papers 1904.07103, arXiv.org, revised Apr 2019.
    7. Fort Gersende & Gobet Emmanuel & Moulines Eric, 2017. "MCMC design-based non-parametric regression for rare event. Application to nested risk computations," Monte Carlo Methods and Applications, De Gruyter, vol. 23(1), pages 21-42, March.

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