Conditional limiting distribution of beta-independent random vectors

The paper deals with random vectors X in R d , d ≥ 2 , possessing the stochastic representation X = d A R V , where R is a positive random radius independent of the random vector V and A ∈ R d × d is a non-singular matrix. If V is uniformly distributed on the unit sphere of R d , then for any integer m < d we have the stochastic representations ( V 1 , … , V m ) = d W ( U 1 , … , U m ) and ( V m + 1 , … , V d ) = d ( 1 − W 2 ) 1 / 2 ( U m + 1 , … , U d ) , with W ≥ 0 , such that W 2 is a beta distributed random variable with parameters m / 2 , ( d − m ) / 2 and ( U 1 , … , U m ) , ( U m + 1 , … , U d ) are independent uniformly distributed on the unit spheres of R m and R d − m , respectively. Assuming a more general stochastic representation for V in this paper we introduce the class of beta-independent random vectors. For this new class we derive several conditional limiting results assuming that R has a distribution function in the max-domain of attraction of a univariate extreme value distribution function. We provide two applications concerning the Kotz approximation of the conditional distributions and the tail asymptotic behaviour of beta-independent bivariate random vectors.