On the infinite divisibility of inverse Beta distributions

Abstract

We show that all negative powers β −s a,b 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 β −s a,b is self-decomposable if and only if 2a + b + s + bs ≥ 1, 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.

A class of scale mixtures of $\operatorname{Gamma}(k)$-distributions that are generalized gamma convolutions

“…Well-known examples of GGC distributions include the log-normal distribution and positive strictly α-stable distributions. Also, each negative power of a gamma variable is shown to have a GGC-distribution in [9]. Bosch and Simon [10] and Jedidi and Simon [14] give other novel results on HM, HCM, and GGC distributions.…”

Density solutions to a class of integro-differential equations

“…The first inclusion X m,α ∈ H ⇒ Y m,α ∈ I of Part (b) is an obvious consequence of (8). As in Theorem 5.1 (b) of [27], the second inclusion Y m,α ∈ I ⇒ m ≤ 2α follows from well-known bounds on the upper tails of positive ID distributions -see e.g.…”

“…where Z ν ρ is the so-called Krätzel function -see (1.7.42) in [22]. On the other hand, we know by Theorem 4 in [8] and the discussion therebefore, that…”

On the law of homogeneous stable functionals

“…By Theorem 4.1.4 in [11], this entails that its Thorin measure has infinite mass, with the terminology of [11]. Using the full extent of Theorem 2.2 in [14], one can also show that all positive powers A(α, 1 − 1/α, q) s with magnitude s ≥ sup(1/2, q/(α + q)) have a GGC distribution with infinite Thorin measure.…”