Beta-gamma tail asymptotics

Understand

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.