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2022 Tail measures and regular variation
Martin Bladt, Enkelejd Hashorva, Georgiy Shevchenko
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Electron. J. Probab. 27: 1-43 (2022). DOI: 10.1214/22-EJP788

Abstract

A general framework for the study of regular variation (RV) is that of Polish star-shaped metric spaces, while recent developments in [41] have discussed RV with respect to a properly localised boundedness B. Along the lines of the latter approach, we discuss the RV of Borel measures and random processes on a general Polish metric spaces (D,dD). Tail measures introduced in [47] appear naturally as limiting measures of regularly varying time series. We define tail measures on the measurable space (D,D) indexed by H(D), a countable family of 1-homogeneous coordinate maps, and show some tractable instances for the investigation of RV when B is determined by H(D). This allows us to study the regular variation of càdlàg processes on D(Rl,Rd) retrieving in particular results obtained in [59] for RV of stationary càdlàg processes on the real line removing l=1 therein. Further, we discuss potential applications and open questions.

Funding Statement

MB kindly acknowledges support from the SNSF Grant 200021-191984 and EH, GS kindly acknowledge support from the SNSF Grant 1200021-196888.

Acknowledgments

We are grateful to both reviewers and the Editor for several comments and suggestions that lead to significant improvement of the manuscript. Many thanks go to Richard Davis, Philippe Soulier, Thomas Mikosch and Ilya Molchanov for literature suggestions and discussions.

Citation

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Martin Bladt. Enkelejd Hashorva. Georgiy Shevchenko. "Tail measures and regular variation." Electron. J. Probab. 27 1 - 43, 2022. https://doi.org/10.1214/22-EJP788

Information

Received: 14 April 2021; Accepted: 27 April 2022; Published: 2022
First available in Project Euclid: 10 May 2022

arXiv: 2103.04396
Digital Object Identifier: 10.1214/22-EJP788

Subjects:
Primary: 28A33 , 60G70

Keywords: càdlàg processes , hidden regular variation , max-stable processes , regular variation , spectral tail processes , tail measures , tail processes , weak convergence

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