The paper deals with random vectors X in R-d, d >= 2, possessing the stochastic representation X (d) double under bar ARV, where R is a positive random radius independent of the random vector V and A is an element of R-dxd is a non-singular matrix. If V is uniformly distributed on the unit sphere of Rd, then for any integer d d m < d we have the stochastic representations (V-1 ,..., V-m) <(d)double under bar> W(U-1,..., U-m) and (Vm+1, V-d) (d) double under bar (1 - w(2))(1/2)(Um+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), (Um+1,..., U-d) are independent uniformly distributed on the unit spheres of R-m and Rd-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.