Asymptotics of the norm of elliptical random vectors

In this paper we consider elliptical random vectors X in Rd,d≥2Rd,d≥2 with stochastic representation ARU, where RR is a positive random radius independent of the random vector U which is uniformly distributed on the unit sphere of RdRd and A∈Rd×dA∈Rd×d is a given matrix. Denote by ‖⋅‖‖⋅‖ the Euclidean norm in RdRd, and let FF be the distribution function of RR. The main result of this paper is an asymptotic expansion of the probability P{‖X‖>u} for FF 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].