Asymptotics of the norm of elliptical random vectors

In this paper we consider elliptical random vectors X in R d , d ≥ 2 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 ∈ R d × d is a given matrix. Denote by ‖ ⋅ ‖ the Euclidean norm in R d , and let F be the distribution function of R . The main result of this paper is an asymptotic expansion of the probability P { ‖ X ‖ > u } for F in the Gumbel or the Weibull max-domain of attraction. In the special case that X is a mean zero Gaussian random vector our result coincides with the one derived in Hüsler et al. (2002) [1].