Multivariate regular variation on cones: application to extreme values, hidden regular variation and conditioned limit laws

Abstract

We attempt to bring some modest unity to three subareas of heavy tail analysis and extreme value theory:

• limit laws for componentwise maxima of iid random variables;

• hidden regular variation and asymptotic independence;

• conditioned limit laws when one component of a random vector is extreme.

The common theme is multivariate regular variation on a cone and the three cases cited come from specifying the cones and [0, ∞] × (0, ∞].

Acknowledgements

Sidney Resnick's research was partially supported by ARO Contract W911NF-07-0078 and NSA Grant H98230-06-1-0069 at Cornell University. Most of the synthesis and writing of this paper took place during May–June 2007 when Sidney Resnick was visiting the Lehrstuhl für Mathematische Statistik, Technical University München and grateful acknowledgement is made for hospitality and support.