Conditional limiting distribution of type III elliptical random vectors (Q864269)

Let \( X =(X_1, \ldots ,X_d)^\top\) be an elliptical random vector in \({\mathcal R}^d\) with stochastic representation \( X =^{\text{ d}}RAU\), where \( U\) is the random vector which is uniformly distributed on the unit sphere of \({\mathcal R}^d\), \(R\) is an almost surely positive random variable with distribution function \(F\) independent of the random vector \( U \), and \(A \in {\mathcal R}^{d \times d}\) is a given non-singular square matrix. Let \(I,J,I \cup J = \{1,\ldots,d\},d \geq 2\), be two non-empty disjoint index sets. Under the assumption that \(F\) is in the Weibull max-domain of attraction, the author shows an asymptotic approximation of the conditional distribution \((X_i, i \in I)^\top \mid ( X_j= a_j, j \in J)^\top,\) \((a_1,\ldots,a_d)^\top \in {\mathcal R}^d\), by letting \( (a_j, j \in J)^\top\) go to some boundary vector.