Asymptotic properties of type I elliptical random vectors

Abstract

Let $ {\boldsymbol X} = A{\boldsymbol S} $ be an elliptical random vector with $ A \in \mathbb{R}^{{k \times k}} ,k \geqslant 2, $ a non-singular square matrix and ${\boldsymbol S}=(S_1, \ldots ,S_k)^\top $ a spherical random vector in $\mathbb{R}^k$ , and let ${\boldsymbol t}_n, n \ge 1$ be a sequence of vectors in $\mathbb{R}^k$ such that $ \lim _{{n \to \infty }} {\boldsymbol P}\{ {\boldsymbol X} > {\boldsymbol 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 ${\boldsymbol P} \{{\boldsymbol X} > {\boldsymbol t}_n\}$ for ${\boldsymbol t}_n$ converging as $ n \to \infty $ 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 ${\boldsymbol X}$ .