On the infinite divisibility of inverse Beta distributions

2014/05/16 by Pierre Bosch, Bosch, Pierre, Thomas Simon +1 · 1 citation
Mathematics · #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical functions and polynomials #Probability (math.PR) #Random Matrices and Applications #math.PR

paper · pdf · doi:10.48550/arxiv.1405.4176

openalex publication_date 2014/05/16 · arxiv created 2014/05/23 · arxiv updated 2014/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Lévy 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.

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