On the infinite divisibility of inverse Beta distributions

Abstract

We show that all negative powers $\beta_{a,b}^{-s}$ of the Beta distribution are infinitely divisible. The case $b\le1$ follows by complete monotonicity, the case $b>1$, $s\ge1$ by hyperbolically complete monotonicity and the case $b>1$, $s<1$ by a Lévy perpetuity argument involving the hypergeometric series. We also observe that $\beta_{a,b}^{-s}$ is self-decomposable if and only if $2a+b+s+bs\ge1$, and that in this case it is not necessarily 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.