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One-component regular variation and graphical modeling of extremes

Part of: Distribution theory - Probability Stochastic processes Multivariate analysis Limit theorems

Published online by Cambridge University Press:  24 October 2016

Adrien Hitz  and
Robin Evans
Adrien Hitz*
Affiliation:
University of Oxford
Robin Evans*
Affiliation:
University of Oxford
*
* Postal address: Department of Statistics, University of Oxford, 1 South Parks Road, Oxford, OX1 3TG, UK.
* Postal address: Department of Statistics, University of Oxford, 1 South Parks Road, Oxford, OX1 3TG, UK.
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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.

Keywords

Regular variationKaramata's theoremhomogeneous distributionmultivariate exceedances over thresholdgraphical model

MSC classification

Primary: 60F99: None of the above, but in this section
Secondary: 60E05: Distributions 62H99: None of the above, but in this section 60G70: Extreme value theory; extremal processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 3 , September 2016 , pp. 733 - 746
Copyright
Copyright © Applied Probability Trust 2016 

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