Asymptotics of the Norm of Elliptical Random Vectors

TL;DR

This paper derives asymptotic expansions for the tail probabilities of the norm of elliptical random vectors, providing insights into their extreme value behavior under specific distributional assumptions.

Contribution

It introduces new asymptotic formulas for the tail probabilities of elliptical vectors' norms when the radius distribution is in the Gumbel or Weibull domain.

Findings

Asymptotic tail probability expansions for elliptical vectors.

Results applicable to Gumbel and Weibull max-domain distributions.

Enhanced understanding of extreme behavior of elliptical random vectors.

Abstract

In this paper we consider elliptical random vectors X in R^d,d>1 with stochastic representation A R U where R is a positive random radius independent of the random vector U which is uniformly distributed on the unit sphere of R^d and A is a given matrix. The main result of this paper is an asymptotic expansion of the tail probability of the norm of X derived under the assumption that R has distribution function is in the Gumbel or the Weibull max-domain of attraction.