Asymptotics of the norm of elliptical random vectors

In this paper we consider elliptical random vectors in with stochastic representation , where R is a positive random radius independent of the random vector which is uniformly distributed on the unit sphere of and is a given matrix. Denote by ||[dot operator]|| the Euclidean norm in , and let F be the distribution function of R. The main result of this paper is an asymptotic expansion of the probability for F in the Gumbel or the Weibull max-domain of attraction. In the special case that is a mean zero Gaussian random vector our result coincides with the one derived in Hüsler et al. (2002) [1].