On the infinite divisibility of inverse Beta distributions

TL;DR

This paper proves that all negative powers of Beta distributions are infinitely divisible, explores their self-decomposability, and shows that negative powers of Gamma distributions are generalized Gamma convolutions, answering a recent open question.

Contribution

It establishes the infinite divisibility of negative powers of Beta distributions using various monotonicity properties and introduces new results about their self-decomposability and relation to generalized Gamma convolutions.

Findings

Negative powers of Beta distributions are infinitely divisible.

Negative powers of Gamma distributions are generalized Gamma convolutions.

Conditions for self-decomposability of Beta distribution powers.

Abstract

We show that all negative powers B_{a,b}^-{s} of the Beta distribution are infinitely divisible. The case b<1 follows by complete monotonicity, the case b > 1, s > 1 by hyperbolically complete monotonicity and the case b > 1, s < 1 by a L\'evy perpetuity argument involving the hypergeometric series. We also observe that B_{a,b}^{-s} is self-decomposable whenever 2a + b + s + bs > 1, and that it is not always a generalized Gamma convolution. On the other hand, we prove that all negative powers of the Gamma distribution are generalized Gamma convolutions, answering to a recent question of L. Bondesson.