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New techniques for empirical processes of dependent data

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  • Dehling, Herold
  • Durieu, Olivier
  • Volny, Dalibor

Abstract

We present a new technique for proving the empirical process invariance principle for stationary processes (Xn)n>=0. The main novelty of our approach lies in the fact that we only require the central limit theorem and a moment bound for a restricted class of functions (f(Xn))n>=0, not containing the indicator functions. Our approach can be applied to Markov chains and dynamical systems, using spectral properties of the transfer operator. Our proof consists of a novel application of chaining techniques.

Suggested Citation

  • Dehling, Herold & Durieu, Olivier & Volny, Dalibor, 2009. "New techniques for empirical processes of dependent data," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3699-3718, October.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:10:p:3699-3718
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    References listed on IDEAS

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    1. Dedecker, Jérôme & Prieur, Clémentine, 2007. "An empirical central limit theorem for dependent sequences," Stochastic Processes and their Applications, Elsevier, vol. 117(1), pages 121-142, January.
    2. Doukhan, Paul & Louhichi, Sana, 1999. "A new weak dependence condition and applications to moment inequalities," Stochastic Processes and their Applications, Elsevier, vol. 84(2), pages 313-342, December.
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    1. Axel Bücher & Holger Dette & Florian Heinrichs, 2020. "Detecting deviations from second-order stationarity in locally stationary functional time series," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(4), pages 1055-1094, August.
    2. Mikosch, Thomas & Zhao, Yuwei, 2015. "The integrated periodogram of a dependent extremal event sequence," Stochastic Processes and their Applications, Elsevier, vol. 125(8), pages 3126-3169.
    3. Dehling, Herold & Durieu, Olivier, 2011. "Empirical processes of multidimensional systems with multiple mixing properties," Stochastic Processes and their Applications, Elsevier, vol. 121(5), pages 1076-1096, May.
    4. Leucht, Anne & Neumann, Michael H., 2013. "Dependent wild bootstrap for degenerate U- and V-statistics," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 257-280.
    5. Damek, Ewa & Mikosch, Thomas & Zhao, Yuwei & Zienkiewicz, Jacek, 2023. "Whittle estimation based on the extremal spectral density of a heavy-tailed random field," Stochastic Processes and their Applications, Elsevier, vol. 155(C), pages 232-267.
    6. Olivier Durieu & Marco Tusche, 2014. "An Empirical Process Central Limit Theorem for Multidimensional Dependent Data," Journal of Theoretical Probability, Springer, vol. 27(1), pages 249-277, March.
    7. Barrera, David & Peligrad, Costel & Peligrad, Magda, 2016. "On the functional CLT for stationary Markov chains started at a point," Stochastic Processes and their Applications, Elsevier, vol. 126(7), pages 1885-1900.
    8. Wang, Yizao, 2014. "Convergence to the maximum process of a fractional Brownian motion with shot noise," Statistics & Probability Letters, Elsevier, vol. 90(C), pages 33-41.
    9. Peligrad, Magda, 2020. "A new CLT for additive functionals of Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5695-5708.

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