Tail asymptotics under beta random scaling

Let X , Y , B be three independent random variables such that X has the same distribution function as YB. Assume that B is a beta random variable with positive parameters α , β and Y has distribution function H with H ( 0 ) = 0 . In this paper we derive a recursive formula for calculation of H, if the distribution function H α , β of X is known. Furthermore, we investigate the relation between the tail asymptotic behaviour of X and Y, which is closely related to asymptotics of Weyl fractional-order integral operators. We present three applications of our asymptotic results concerning the extremes of two random samples with underlying distribution functions H and H α , β , respectively, and the conditional limiting distribution of bivariate elliptical distributions.