  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 a,b and Y has distribution function H. Pakes and Navarro
(2007) show under some mild conditions that the distribution function H_{a,b}
of X determines H. Based on that result we derive in this paper a recursive
formula for calculation of H, if H_{a,b} 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 H_{a,b}, respectively, and
the conditional limiting distribution of bivariate elliptical distributions.
