Journal of Applied Probability

One-component regular variation and graphical modeling of extremes

Adrien Hitz and Robin Evans

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Abstract

The problem of inferring the distribution of a random vector given that its norm is large requires modeling a homogeneous limiting density. We suggest an approach based on graphical models which is suitable for high-dimensional vectors. We introduce the notion of one-component regular variation to describe a function that is regularly varying in its first component. We extend the representation and Karamata's theorem to one-component regularly varying functions, probability distributions and densities, and explain why these results are fundamental in multivariate extreme-value theory. We then generalize the Hammersley–Clifford theorem to relate asymptotic conditional independence to a factorization of the limiting density, and use it to model multivariate tails.

Article information

Source
J. Appl. Probab., Volume 53, Number 3 (2016), 733-746.

Dates
First available in Project Euclid: 13 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.jap/1476370773

Mathematical Reviews number (MathSciNet)
MR3570091

Zentralblatt MATH identifier
1353.60036

Subjects
Primary: 60F99: None of the above, but in this section
Secondary: 60E05: Distributions: general theory 62H99: None of the above, but in this section 60G70: Extreme value theory; extremal processes

Keywords
Regular variation Karamata's theorem homogeneous distribution multivariate exceedances over threshold graphical model

Citation

Hitz, Adrien; Evans, Robin. One-component regular variation and graphical modeling of extremes. J. Appl. Probab. 53 (2016), no. 3, 733--746. https://projecteuclid.org/euclid.jap/1476370773


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