Let X = AS be an elliptical random vector with A ∈ R~(k×k), k ≥ 2, a non-singular square matrix andS = (S_1,...,S_k)~T a spherical random vector in R~k, and let t_n, n ≥ 1 be a sequence of vectors in R~k such that lim_(n→∞) P{X > t_n} = 0. We assume in this paper that the associated random radius R_k = (S_1 + S_2 + ... + S_k)~(1/2) is almost surely positive, and it has distribution function in the Gumbel max-domain of attraction. Relying on extreme value theory we obtain an exact asymptotic expansion of the tail probability P{X > t_n} for t_n converging as n → ∞ to a boundary point. Further we discuss density convergence under a suitable transformation. We apply our results to obtain an asymptotic approximation of the distribution of partial excess above a high threshold, and to derive a conditional limiting result. Further, we investigate the asymptotic behaviour of concomitants of order statistics, and the tail asymptotics of associated random radius for subvectors of X.
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