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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
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    References listed on IDEAS

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    1. 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.
    2. Steinwart, Ingo & Hush, Don & Scovel, Clint, 2009. "Learning from dependent observations," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 175-194, January.
    3. Hang, H. & Steinwart, I., 2014. "Fast learning from α-mixing observations," Journal of Multivariate Analysis, Elsevier, vol. 127(C), pages 184-199.
    4. 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.
    5. 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.
    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.
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    Cited by:

    1. Li, Mengbing & Shi, Chengchun & Wu, Zhenke & Fryzlewicz, Piotr, 2025. "Testing stationarity and change point detection in reinforcement learning," LSE Research Online Documents on Economics 127507, London School of Economics and Political Science, LSE Library.

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