IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v152y2022icp452-485.html
   My bibliography  Save this article

Deviation inequalities for stochastic approximation by averaging

Author

Listed:
  • Fan, Xiequan
  • Alquier, Pierre
  • Doukhan, Paul

Abstract

We introduce a class of Markov chains that includes models of stochastic approximation by averaging and non-averaging. Using a martingale approximation method, we establish various deviation inequalities for separately Lipschitz functions of such a chain, with different moment conditions on some dominating random variables of martingale differences. Finally, we apply these inequalities to stochastic approximation by averaging and empirical risk minimization.

Suggested Citation

  • Fan, Xiequan & Alquier, Pierre & Doukhan, Paul, 2022. "Deviation inequalities for stochastic approximation by averaging," Stochastic Processes and their Applications, Elsevier, vol. 152(C), pages 452-485.
    • Handle: RePEc:eee:spapps:v:152:y:2022:i:c:p:452-485
    DOI: 10.1016/j.spa.2022.07.002
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414922001600
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2022.07.002?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 search for a different version of it.

    References listed on IDEAS

    as
    1. Lesigne, Emmanuel & Volný, Dalibor, 2001. "Large deviations for martingales," Stochastic Processes and their Applications, Elsevier, vol. 96(1), pages 143-159, November.
    2. Phillips, Peter C B, 1988. "Regression Theory for Near-Integrated Time Series," Econometrica, Econometric Society, vol. 56(5), pages 1021-1043, September.
    3. Alquier Pierre & Li Xiaoyin & Wintenberger Olivier, 2014. "Prediction of time series by statistical learning: general losses and fast rates," Dependence Modeling, De Gruyter, vol. 1(2013), pages 65-93, January.
    4. Dedecker, Jérôme & Fan, Xiequan, 2015. "Deviation inequalities for separately Lipschitz functionals of iterated random functions," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 60-90.
    5. Phillips, Peter C.B. & Magdalinos, Tassos, 2007. "Limit theory for moderate deviations from a unit root," Journal of Econometrics, Elsevier, vol. 136(1), pages 115-130, January.
    6. Hang, H. & Steinwart, I., 2014. "Fast learning from α-mixing observations," Journal of Multivariate Analysis, Elsevier, vol. 127(C), pages 184-199.
    7. Emmanuel Rio, 2009. "Moment Inequalities for Sums of Dependent Random Variables under Projective Conditions," Journal of Theoretical Probability, Springer, vol. 22(1), pages 146-163, March.
    8. Fan, Xiequan & Grama, Ion & Liu, Quansheng, 2012. "Hoeffding’s inequality for supermartingales," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3545-3559.
    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. Alquier Pierre & Doukhan Paul & Fan Xiequan, 2019. "Exponential inequalities for nonstationary Markov chains," Dependence Modeling, De Gruyter, vol. 7(1), pages 150-168, January.
    2. Ying Wang & Peter C. B. Phillips & Yundong Tu, 2024. "Limit Theory and Inference in Non-cointegrated Functional Coefficient Regression," Cowles Foundation Discussion Papers 2399, Cowles Foundation for Research in Economics, Yale University.
    3. Bykhovskaya, Anna & Phillips, Peter C.B., 2020. "Point optimal testing with roots that are functionally local to unity," Journal of Econometrics, Elsevier, vol. 219(2), pages 231-259.
    4. HORIE, Tetsushi & 堀江, 哲史 & YAMAMOTO, Yohei & 山本, 庸平, 2016. "Testing for Speculative Bubbles in Large-Dimensional Financial Panel Data Sets," Discussion Papers 2016-04, Graduate School of Economics, Hitotsubashi University.
    5. Gonzalo, Jesus & Pitarakis, Jean-Yves, 2010. "Regime specific predictability in predictive regressions," Discussion Paper Series In Economics And Econometrics 0916, Economics Division, School of Social Sciences, University of Southampton.
    6. Horváth, Lajos & Trapani, Lorenzo, 2019. "Testing for randomness in a random coefficient autoregression model," Journal of Econometrics, Elsevier, vol. 209(2), pages 338-352.
    7. Jardet, Caroline & Monfort, Alain & Pegoraro, Fulvio, 2013. "No-arbitrage Near-Cointegrated VAR(p) term structure models, term premia and GDP growth," Journal of Banking & Finance, Elsevier, vol. 37(2), pages 389-402.
    8. Jesús Gonzalo & Jean-Yves Pitarakis, 2011. "Regime-Specific Predictability in Predictive Regressions," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 30(2), pages 229-241, June.
    9. Ibragimov, Rustam & Walden, Johan, 2008. "Portfolio diversification under local and moderate deviations from power laws," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 594-599, April.
    10. Nannan Ma & Hailin Sang & Guangyu Yang, 2023. "Least absolute deviation estimation for AR(1) processes with roots close to unity," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 75(5), pages 799-832, October.
    11. Dedecker, Jérôme & Fan, Xiequan, 2015. "Deviation inequalities for separately Lipschitz functionals of iterated random functions," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 60-90.
    12. Christis Katsouris, 2023. "Quantile Time Series Regression Models Revisited," Papers 2308.06617, arXiv.org, revised Aug 2023.
    13. Yijie Fei & Yiu Lim Lui & Jun Yu, 2024. "Testing Predictability in the Presence of Persistent Errors," Working Papers 202401, University of Macau, Faculty of Business Administration.
    14. Christis Katsouris, 2023. "Limit Theory under Network Dependence and Nonstationarity," Papers 2308.01418, arXiv.org, revised Aug 2023.
    15. Phillips, Peter C.B. & Lee, Ji Hyung, 2016. "Robust econometric inference with mixed integrated and mildly explosive regressors," Journal of Econometrics, Elsevier, vol. 192(2), pages 433-450.
    16. Jeong, Minsoo, 2022. "Modelling persistent stationary processes in continuous time," Economic Modelling, Elsevier, vol. 109(C).
    17. Sabzikar, Farzad & Wang, Qiying & Phillips, Peter C.B., 2020. "Asymptotic theory for near integrated processes driven by tempered linear processes," Journal of Econometrics, Elsevier, vol. 216(1), pages 192-202.
    18. Christis Katsouris, 2023. "Structural Break Detection in Quantile Predictive Regression Models with Persistent Covariates," Papers 2302.05193, arXiv.org.
    19. Christis Katsouris, 2023. "Bootstrapping Nonstationary Autoregressive Processes with Predictive Regression Models," Papers 2307.14463, arXiv.org.
    20. Peter C.B. Phillips & Tassos Magadalinos, 2005. "Limit Theory for Moderate Deviations from a Unit Root under Weak Dependence," Cowles Foundation Discussion Papers 1517, Cowles Foundation for Research in Economics, Yale University.

    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:152:y:2022:i:c:p:452-485. 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.