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Spectral tail processes and max-stable approximations of multivariate regularly varying time series

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  • Janßen, Anja

Abstract

A regularly varying time series as introduced in Basrak and Segers (2009) is a (multivariate) time series such that all finite dimensional distributions are multivariate regularly varying. The extremal behavior of such a process can then be described by the index of regular variation and the so-called spectral tail process, which is the limiting distribution of the rescaled process, given an extreme event at time 0. As shown in Basrak and Segers (2009), the stationarity of the underlying time series implies a certain structure of the spectral tail process, informally known as the “time change formula”. In this article, we show that on the other hand, every process which satisfies this property is in fact the spectral tail process of an underlying stationary max-stable process. The spectral tail process and the corresponding max-stable process then provide two complementary views on the extremal behavior of a multivariate regularly varying stationary time series.

Suggested Citation

  • Janßen, Anja, 2019. "Spectral tail processes and max-stable approximations of multivariate regularly varying time series," Stochastic Processes and their Applications, Elsevier, vol. 129(6), pages 1993-2009.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:6:p:1993-2009
    DOI: 10.1016/j.spa.2018.06.010
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    References listed on IDEAS

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    1. Segers, Johan & Zhao, Yuwei & Meinguet, Thomas, 2017. "Polar decomposition of regularly varying time series in star-shaped metric spaces," LIDAM Reprints ISBA 2017029, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Dombry, Clément & Kabluchko, Zakhar, 2017. "Ergodic decompositions of stationary max-stable processes in terms of their spectral functions," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1763-1784.
    3. Basrak, Bojan & Davis, Richard A. & Mikosch, Thomas, 2002. "Regular variation of GARCH processes," Stochastic Processes and their Applications, Elsevier, vol. 99(1), pages 95-115, May.
    4. 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:

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    2. Buriticá, Gloria & Mikosch, Thomas & Wintenberger, Olivier, 2023. "Large deviations of ℓp-blocks of regularly varying time series and applications to cluster inference," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 68-101.

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