Beta-gamma tail asymptotics

TL;DR

This paper derives the tail asymptotics for the product of independent beta and generalized gamma random variables, simplifying previous proofs and aiding in understanding the maximum degree in preferential attachment trees.

Contribution

It provides a simpler proof for the tail asymptotics of the product of beta and generalized gamma variables, extending previous results and applications.

Findings

Derived explicit tail asymptotics for the product of beta and generalized gamma variables

Simplified the proof of these asymptotics compared to previous work

Applied results to analyze the maximum degree in preferential attachment trees

Abstract

We compute the tail asymptotics of the product of a beta random variable and a generalized gamma random variable which are independent and have general parameters. A special case of these asymptotics were proved and used in a recent work of Bubeck, Mossel, and R\'acz in order to determine the tail asymptotics of the maximum degree of the preferential attachment tree. The proof presented here is simpler and highlights why these asymptotics hold.