On Beta-Product Convolutions

TL;DR

This paper explores the relationship between the distribution of a positive random variable R and its product with a beta-distributed variable S, focusing on tail asymptotics and applications in elliptical and polar distributions.

Contribution

It derives new distributional properties and tail asymptotics for the product RS, extending understanding of their behavior in elliptical and polar distribution models.

Findings

Characterizes the tail behavior of RS when R is regularly varying at 0.

Provides asymptotic analysis of componentwise sample minima in elliptical distributions.

Analyzes the lower tails of aggregated risk in bivariate polar distributions.

Abstract

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. Our first application concerns the asymptotic behaviour of the componentwise sample minima related to an elliptical distributions. Further, we derive the lower tails asymptotic of the aggregated risk for bivariate polar distributions.