IDEAS home Printed from https://ideas.repec.org/a/vrs/demode/v7y2019i1p150-168n7.html
   My bibliography  Save this article

Exponential inequalities for nonstationary Markov chains

Author

Listed:
  • Alquier Pierre

    (CREST, ENSAE, Université Paris Saclay)

  • Doukhan Paul

    (AGM UMR8088 UniversityParis-Seine and CIMFAV, Universidad de Valparaiso, Chile)

  • Fan Xiequan

    (CAM, Tianjin University, Tianjin, China)

Abstract

Exponential inequalities are main tools in machine learning theory. To prove exponential inequalities for non i.i.d random variables allows to extend many learning techniques to these variables. Indeed, much work has been done both on inequalities and learning theory for time series, in the past 15 years. However, for the non independent case, almost all the results concern stationary time series. This excludes many important applications: for example any series with a periodic behaviour is nonstationary. In this paper, we extend the basic tools of [19] to nonstationary Markov chains. As an application, we provide a Bernsteintype inequality, and we deduce risk bounds for the prediction of periodic autoregressive processes with an unknown period.

Suggested Citation

  • Alquier Pierre & Doukhan Paul & Fan Xiequan, 2019. "Exponential inequalities for nonstationary Markov chains," Dependence Modeling, De Gruyter, vol. 7(1), pages 150-168, January.
  • Handle: RePEc:vrs:demode:v:7:y:2019:i:1:p:150-168:n:7
    DOI: 10.1515/demo-2019-0007
    as

    Download full text from publisher

    File URL: https://doi.org/10.1515/demo-2019-0007
    Download Restriction: no

    File URL: https://libkey.io/10.1515/demo-2019-0007?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
    ---><---

    References listed on IDEAS

    as
    1. Steinwart, Ingo & Hush, Don & Scovel, Clint, 2009. "Learning from dependent observations," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 175-194, January.
    2. Hang, H. & Steinwart, I., 2014. "Fast learning from α-mixing observations," Journal of Multivariate Analysis, Elsevier, vol. 127(C), pages 184-199.
    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. Dahlhaus, R., 1996. "On the Kullback-Leibler information divergence of locally stationary processes," Stochastic Processes and their Applications, Elsevier, vol. 62(1), pages 139-168, March.
    5. 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.
    6. Doukhan, Paul & Neumann, Michael H., 2007. "Probability and moment inequalities for sums of weakly dependent random variables, with applications," Stochastic Processes and their Applications, Elsevier, vol. 117(7), pages 878-903, July.
    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. 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.
    2. Chen, Qitong & Hong, Yongmiao & Li, Haiqi, 2024. "Time-varying forecast combination for factor-augmented regressions with smooth structural changes," Journal of Econometrics, Elsevier, vol. 240(1).
    3. Philippe Goulet Coulombe & Maxime Leroux & Dalibor Stevanovic & Stéphane Surprenant, 2022. "How is machine learning useful for macroeconomic forecasting?," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 37(5), pages 920-964, August.
    4. Fryzlewicz, Piotr & Nason, Guy P., 2006. "Haar-Fisz estimation of evolutionary wavelet spectra," LSE Research Online Documents on Economics 25227, London School of Economics and Political Science, LSE Library.
    5. Hafouta, Yeor, 2023. "Convergence rates in the functional CLT for α-mixing triangular arrays," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 242-290.
    6. Yayi Yan & Jiti Gao & Bin Peng, 2021. "On Time-Varying VAR Models: Estimation, Testing and Impulse Response Analysis," Papers 2111.00450, arXiv.org.
    7. Aloy Marcel & Dufrénot Gilles & Tong Charles Lai & Peguin-Feissolle Anne, 2013. "A smooth transition long-memory model," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 17(3), pages 281-296, May.
    8. Baruník, Jozef & Ellington, Michael, 2024. "Persistence in financial connectedness and systemic risk," European Journal of Operational Research, Elsevier, vol. 314(1), pages 393-407.
    9. Dew-Becker, Ian & Nathanson, Charles G., 2019. "Directed attention and nonparametric learning," Journal of Economic Theory, Elsevier, vol. 181(C), pages 461-496.
    10. Fryzlewicz, Piotr & Nason, Guy P., 2004. "Smoothing the wavelet periodogram using the Haar-Fisz transform," LSE Research Online Documents on Economics 25231, London School of Economics and Political Science, LSE Library.
    11. Beran, Jan, 2007. "On parameter estimation for locally stationary long-memory processes," CoFE Discussion Papers 07/13, University of Konstanz, Center of Finance and Econometrics (CoFE).
    12. Hwang, Eunju & Shin, Dong Wan, 2012. "Strong consistency of the stationary bootstrap under ψ-weak dependence," Statistics & Probability Letters, Elsevier, vol. 82(3), pages 488-495.
    13. Stefan Birr & Stanislav Volgushev & Tobias Kley & Holger Dette & Marc Hallin, 2017. "Quantile spectral analysis for locally stationary time series," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(5), pages 1619-1643, November.
    14. Arif Dowla & Efstathios Paparoditis & Dimitris Politis, 2013. "Local block bootstrap inference for trending time series," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(6), pages 733-764, August.
    15. Eunju Hwang & Dong Shin, 2016. "Kernel estimators of mode under $$\psi $$ ψ -weak dependence," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 68(2), pages 301-327, April.
    16. Roueff, François & von Sachs, Rainer & Sansonnet, Laure, 2016. "Locally stationary Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1710-1743.
    17. Jiti Gao & Bin Peng & Wei Biao Wu & Yayi Yan, 2022. "Time-Varying Multivariate Causal Processes," Monash Econometrics and Business Statistics Working Papers 8/22, Monash University, Department of Econometrics and Business Statistics.
    18. Last, Michael & Shumway, Robert, 2008. "Detecting abrupt changes in a piecewise locally stationary time series," Journal of Multivariate Analysis, Elsevier, vol. 99(2), pages 191-214, February.
    19. Jozef Barunik & Michael Ellington, 2020. "Dynamic Network Risk," Papers 2006.04639, arXiv.org, revised Jul 2020.
    20. Matteo Barigozzi & Christian Brownlees, 2019. "NETS: Network estimation for time series," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 34(3), pages 347-364, April.

    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:vrs:demode:v:7:y:2019:i:1:p:150-168:n:7. 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: Peter Golla (email available below). General contact details of provider: https://www.degruyter.com .

    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.