Conditional limiting distribution of Type III elliptical random vectors

In this paper we consider elliptical random vectors in R d , d ≥ 2 with stochastic representation RA U where R is a positive random radius independent of the random vector U which is uniformly distributed on the unit sphere of R d and A ∈ R d × d is a non-singular matrix. When R has distribution function in the Weibull max-domain of attraction we say that the corresponding elliptical random vector is of Type III. For the bivariate set-up, Berman [Sojurns and Extremes of Stochastic Processes, Wadsworth & Brooks/ Cole, 1992] obtained for Type III elliptical random vectors an interesting asymptotic approximation by conditioning on one component. In this paper we extend Bermanʹs result to Type III elliptical random vectors in R d . Further, we derive an asymptotic approximation for the conditional distribution of such random vectors.