On Beta-Product Convolutions

Let R be a positive random variable independent of S which is beta distributed. In this paper we are interested on the relation between the distribution function of $R$ and that of RS. For this model we derive first some distributional properties, and then investigate the lower tail asymptotics of RS when R is regularly varying at 0, and vice-versa. The applications we present in this paper concern a) the simplicity of Dirichlet distributions, b) asymptotics of the sample minima of elliptical distributions, and c) the effect of the scaling on the asymptotics of aggregated risks.