One-component regular variation and graphical modeling of extremes. (English) Zbl 1353.60036

Hitz, Adrien; Evans, Robin

J. Appl. Probab. 53, No. 3, 733-746 (2016).

Summary: 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.

MSC:

60F99	Limit theorems in probability theory
60G70	Extreme value theory; extremal stochastic processes
60E05	Probability distributions: general theory
62H99	Multivariate analysis

Keywords:

regular variation; extremes; Karamata’s theorem; homogeneous distribution; multivariate exceedances; graphical model