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Polar decomposition of regularly varying time series in star-shaped metric spaces

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Abstract

There exist two ways of defining regular variation of a time series in a star-shaped metric space: either by the distributions of finite stretches of the series or by viewing the whole series as a single random element in a sequence space. The two definitions are shown to be equivalent. The introduction of a norm-like function, called modulus, yields a polar decomposition similar to the one in Euclidean spaces. The angular component of the time series, called angular or spectral tail process, captures all aspects of extremal dependence. The stationarity of the underlying series induces a transformation formula of the spectral tail process under time shifts.

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Correspondence to Yuwei Zhao.

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The first author’s research is supported by contract nr. 07/12-045 of the Projet d’Actions de Recherche Concerté es of the Communauté française de Belgique, granted by the Académie universitaire Louvain and the second author’s research is supported by IAP research network grant nr. P7/06 of the Belgian government (Belgian Science Policy).

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Segers, J., Zhao, Y. & Meinguet, T. Polar decomposition of regularly varying time series in star-shaped metric spaces. Extremes 20, 539–566 (2017). https://doi.org/10.1007/s10687-017-0287-3

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  • DOI: https://doi.org/10.1007/s10687-017-0287-3

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