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Bounds on regeneration times and convergence rates for Markov chains

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  • Roberts, G. O.
  • Tweedie, R. L.

Abstract

In many applications of Markov chains, and especially in Markov chain Monte Carlo algorithms, the rate of convergence of the chain is of critical importance. Most techniques to establish such rates require bounds on the distribution of the random regeneration time T that can be constructed, via splitting techniques, at times of return to a "small set" C satisfying a minorisation condition P(x,·)[greater-or-equal, slanted][var epsilon][phi](·), x[set membership, variant]C. Typically, however, it is much easier to get bounds on the time [tau]C of return to the small set itself, usually based on a geometric drift function , where . We develop a new relationship between T and [tau]C, and this gives a bound on the tail of T, based on [var epsilon],[lambda] and b, which is a strict improvement on existing results. When evaluating rates of convergence we see that our bound usually gives considerable numerical improvement on previous expressions.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:80:y:1999:i:2:p:211-229
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    References listed on IDEAS

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    1. 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. Svetlana Ekisheva & Mark Borodovsky, 2011. "Uniform Accuracy of the Maximum Likelihood Estimates for Probabilistic Models of Biological Sequences," Methodology and Computing in Applied Probability, Springer, vol. 13(1), pages 105-120, March.
    2. Jarner, S. F. & Tweedie, R. L., 2002. "Convergence rates and moments of Markov chains associated with the mean of Dirichlet processes," Stochastic Processes and their Applications, Elsevier, vol. 101(2), pages 257-271, October.
    3. Roberts, Gareth O. & Rosenthal, Jeffrey S., 2002. "One-shot coupling for certain stochastic recursive sequences," Stochastic Processes and their Applications, Elsevier, vol. 99(2), pages 195-208, June.
    4. Quan Zhou & Jun Yang & Dootika Vats & Gareth O. Roberts & Jeffrey S. Rosenthal, 2022. "Dimension‐free mixing for high‐dimensional Bayesian variable selection," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(5), pages 1751-1784, November.
    5. Athreya, Krishna B. & Roy, Vivekananda, 2014. "When is a Markov chain regenerative?," Statistics & Probability Letters, Elsevier, vol. 84(C), pages 22-26.
    6. Yang, Jun & Roberts, Gareth O. & Rosenthal, Jeffrey S., 2020. "Optimal scaling of random-walk metropolis algorithms on general target distributions," Stochastic Processes and their Applications, Elsevier, vol. 130(10), pages 6094-6132.
    7. Jarner, Søren Fiig & Hansen, Ernst, 2000. "Geometric ergodicity of Metropolis algorithms," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 341-361, February.
    8. Mattingly, J. C. & Stuart, A. M. & Higham, D. J., 2002. "Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise," Stochastic Processes and their Applications, Elsevier, vol. 101(2), pages 185-232, October.
    9. Jeffrey S. Rosenthal, 2003. "Geometric Convergence Rates for Time-Sampled Markov Chains," Journal of Theoretical Probability, Springer, vol. 16(3), pages 671-688, July.
    10. 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.
    11. Gareth O. Roberts & Jeffrey S. Rosenthal, 2019. "Hitting Time and Convergence Rate Bounds for Symmetric Langevin Diffusions," Methodology and Computing in Applied Probability, Springer, vol. 21(3), pages 921-929, September.
    12. Fort, G. & Moulines, E., 2003. "Polynomial ergodicity of Markov transition kernels," Stochastic Processes and their Applications, Elsevier, vol. 103(1), pages 57-99, January.

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