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The tail empirical process of regularly varying functions of geometrically ergodic Markov chains

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  • Kulik, Rafał
  • Soulier, Philippe
  • Wintenberger, Olivier

Abstract

We consider a stationary regularly varying time series which can be expressed as a function of a geometrically ergodic Markov chain. We obtain practical conditions for the weak convergence of the tail array sums and feasible estimators of cluster statistics. These conditions include the so-called geometric drift or Foster–Lyapunov condition and can be easily checked for most usual time series models with a Markovian structure. We illustrate these conditions on several models and statistical applications. A counterexample is given to show a different limiting behavior when the geometric drift condition is not fulfilled.

Suggested Citation

  • Kulik, Rafał & Soulier, Philippe & Wintenberger, Olivier, 2019. "The tail empirical process of regularly varying functions of geometrically ergodic Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4209-4238.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:11:p:4209-4238
    DOI: 10.1016/j.spa.2018.11.014
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    1. Basrak, Bojan & Segers, Johan, 2009. "Regularly varying multivariate time series," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1055-1080, April.
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    Cited by:

    1. Drees, Holger & Janßen, Anja & Neblung, Sebastian, 2021. "Cluster based inference for extremes of time series," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 1-33.
    2. Bücher, Axel & Jennessen, Tobias, 2022. "Statistical analysis for stationary time series at extreme levels: New estimators for the limiting cluster size distribution," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 75-106.
    3. Rasmus Pedersen & Olivier Wintenberger, 2017. "On the tail behavior of a class of multivariate conditionally heteroskedastic processes," Papers 1701.05091, arXiv.org, revised Dec 2017.

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