Extremes of conditioned elliptical random vectors

Let {Xn,n⩾1} be iid elliptical random vectors in Rd,d≥2 and let I,J be two non-empty disjoint index sets. Denote by Xn,I,Xn,J the subvectors of Xn with indices in I,J, respectively. For any a∈Rd such that aJ is in the support of X1,J the conditional random sample Xn,I|Xn,J=aJ,n≥1 consists of elliptically distributed random vectors. In this paper we investigate the relation between the asymptotic behaviour of the multivariate extremes of the conditional sample and the unconditional one. We show that the asymptotic behaviour of the multivariate extremes of both samples is the same, provided that the associated random radius of X1 has distribution function in the max-domain of attraction of a univariate extreme value distribution.