IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v196y2026ics0304414926000372.html

Almost sure convergence rates of adaptive increasingly rare Markov chain Monte Carlo

Author

Listed:
  • Hofstadler, Julian
  • Łatuszyński, Krzysztof
  • Roberts, Gareth
  • Rudolf, Daniel

Abstract

We consider adaptive increasingly rare Markov chain Monte Carlo (MCMC) algorithms, which are adaptive MCMC methods, where the adaptation concerning the “past” happens less and less frequently over time. Under a contraction assumption with respect to a Wasserstein-like function we deduce upper bounds of the convergence rate of Monte Carlo sums taking a renormalisation factor into account that is “almost” the one that appears in a law of the iterated logarithm. We demonstrate the applicability of our results by considering different settings, among which are those of simultaneous geometric and uniform ergodicity. All proofs are carried out on an augmented state space, including the classical non-augmented setting as a special case. In contrast to other adaptive MCMC limit theory, some technical assumptions, like diminishing adaptation, are not needed.

Suggested Citation

  • Hofstadler, Julian & Łatuszyński, Krzysztof & Roberts, Gareth & Rudolf, Daniel, 2026. "Almost sure convergence rates of adaptive increasingly rare Markov chain Monte Carlo," Stochastic Processes and their Applications, Elsevier, vol. 196(C).
    • Handle: RePEc:eee:spapps:v:196:y:2026:i:c:s0304414926000372
    DOI: 10.1016/j.spa.2026.104905
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414926000372
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2026.104905?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Glynn, Peter W. & Ormoneit, Dirk, 2002. "Hoeffding's inequality for uniformly ergodic Markov chains," Statistics & Probability Letters, Elsevier, vol. 56(2), pages 143-146, January.
    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. Miasojedow, Błażej, 2014. "Hoeffding’s inequalities for geometrically ergodic Markov chains on general state space," Statistics & Probability Letters, Elsevier, vol. 87(C), pages 115-120.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Sandrić, Nikola & Šebek, Stjepan, 2023. "Hoeffding’s inequality for non-irreducible Markov models," Statistics & Probability Letters, Elsevier, vol. 200(C).
    2. Fort, G. & Moulines, E., 2003. "Polynomial ergodicity of Markov transition kernels," Stochastic Processes and their Applications, Elsevier, vol. 103(1), pages 57-99, January.
    3. H. S. Chang, 2004. "Technical Note: On Ordinal Comparison of Policies in Markov Reward Processes," Journal of Optimization Theory and Applications, Springer, vol. 122(1), pages 207-217, July.
    4. Chris Sherlock & Anthony Lee, 2022. "Variance Bounding of Delayed-Acceptance Kernels," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 2237-2260, September.
    5. Koch, Erwan & Robert, Christian Y., 2019. "Geometric ergodicity for some space–time max-stable Markov chains," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 43-49.
    6. Dalalyan, Arnak S. & Karagulyan, Avetik, 2019. "User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 5278-5311.
    7. Yongqiang Tang, 2007. "A Hoeffding-Type Inequality for Ergodic Time Series," Journal of Theoretical Probability, Springer, vol. 20(2), pages 167-176, June.
    8. Arnak S. Dalalyan, 2017. "Theoretical guarantees for approximate sampling from smooth and log-concave densities," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(3), pages 651-676, June.
    9. O. F. Christensen & J. Møller & R. P. Waagepetersen, 2001. "Geometric Ergodicity of Metropolis-Hastings Algorithms for Conditional Simulation in Generalized Linear Mixed Models," Methodology and Computing in Applied Probability, Springer, vol. 3(3), pages 309-327, September.
    10. Flore Sentenac & Nathan Noiry & Matthieu Lerasle & Laurent Ménard & Vianney Perchet, 2026. "Online Matching in Geometric Random Graphs," Mathematics of Operations Research, INFORMS, vol. 51(2), pages 1585-1625, May.
    11. Fan, Jianqing & Guo, Yongyi & Jiang, Bai, 2022. "Adaptive Huber regression on Markov-dependent data," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 802-818.
    12. Ahmad, I.A. & Amezziane, M., 2013. "Probability inequalities for bounded random vectors," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 1136-1142.
    13. Penev, Spiridon & Peng, Hanxiang & Schick, Anton & Wefelmeyer, Wolfgang, 2004. "Efficient estimators for functionals of Markov chains with parametric marginals," Statistics & Probability Letters, Elsevier, vol. 66(3), pages 335-345, February.
    14. Allassonnière, Stéphanie & Kuhn, Estelle, 2015. "Convergent stochastic Expectation Maximization algorithm with efficient sampling in high dimension. Application to deformable template model estimation," Computational Statistics & Data Analysis, Elsevier, vol. 91(C), pages 4-19.
    15. Liu, Jinpeng & Liu, Yuanyuan & Zhao, Yiqiang Q., 2022. "Augmented truncation approximations to the solution of Poisson’s equation for Markov chains," Applied Mathematics and Computation, Elsevier, vol. 414(C).
    16. Lacour, Claire, 2008. "Nonparametric estimation of the stationary density and the transition density of a Markov chain," Stochastic Processes and their Applications, Elsevier, vol. 118(2), pages 232-260, February.
    17. Denis Belomestny & Leonid Iosipoi, 2019. "Fourier transform MCMC, heavy tailed distributions and geometric ergodicity," Papers 1909.00698, arXiv.org, revised Dec 2019.
    18. Fort, Gersende & Moulines, Eric, 2000. "V-Subgeometric ergodicity for a Hastings-Metropolis algorithm," Statistics & Probability Letters, Elsevier, vol. 49(4), pages 401-410, October.
    19. Belomestny, Denis & Iosipoi, Leonid, 2021. "Fourier transform MCMC, heavy-tailed distributions, and geometric ergodicity," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 181(C), pages 351-363.
    20. Shie Mannor & John N. Tsitsiklis, 2005. "On the Empirical State-Action Frequencies in Markov Decision Processes Under General Policies," Mathematics of Operations Research, INFORMS, vol. 30(3), pages 545-561, August.

    More about this item

    Keywords

    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:196:y:2026:i:c:s0304414926000372. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.