Tail Asymptotics under Beta Random Scaling

Abstract: Let X,Y,B be three independent random variables such that X has the same distribution function as Y B. Assume that B is a beta random variable with positive parameters α, β and Y has distribution function H with H(0) = 0. Pakes and Navarro (2007) show under some mild conditions that the distribution function Hα,β of X determines H . Based on that result we derive in this paper a recursive formula for calculation of H , if Hα,β is known. Furthermore, we investigate the relation between the tail asymptotic behaviour of X and Y . We present three applications of our asymptotic results concerning the extremes of two random samples with underlying distribution functions H and Hα,β , respectively, and the conditional limiting distribution of bivariate elliptical distributions.