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

Abstract

Let k > 0 be an integer and Y a standard Gamma(k) distributed random variable. Let X be an independent positive random variable with a density that is hyperbolically monotone (HM) of order k. Then Y ·X and Y /X both have distributions that are generalized gamma convolutions (GGCs). This result extends a result of Roynette et al. from 2009 who treated the case k = 1 but without use of the HM-concept. Applications in excursion theory of diffusions and in the theory of exponential functionals of Lévy processes are mentioned.2010 Mathematics subject classification. 60E05, 60E10 (primary), 60G17, 60G51, 60J60 (secondary)

Stieltjes functions of finite order and hyperbolic monotonicity

“…as x → 0, by Proposition 2 applied toĝ. This yields g (2) (∞) = 0 and clearly, we have g (i) (∞) = 0 as well for every i = 3, . .…”

“…In this paper however, we will stay within the realm of functions of one real variable. It is plain by dominated convergence that a function f ∈ S is also completely monotone (f ∈ CM for short), in other words f is smooth and (−1) n f (n) ≥ 0 (2) for all n ≥ 0 where, here and throughout, f (n) stands for the n−th derivative of f. Recall from Bernstein's theorem -see e.g. Theorem 1.4 in [10] -that f ∈ CM if and only if there exists a non-negative measure µ(dt) on [0, ∞) (the so-called Bernstein measure) such that…”

“…If k = 2, then ϕ ′ 2 is a negative measure, which implies ϕ 2 (0+) > −∞. Supposing ϕ 2 (0+) > 0, then recalling ϕ 2 = x 3 g (2) , we see by integration that g ′ (0+) = −∞. This is a contradiction since g is non-decreasing, by the fact that f also belongs to S 1 .…”

On Mixtures of Gamma distributions, distributions with hyperbolically monotone densities and Generalized Gamma Convolutions (GGC)

“…Roynette et al [5] proved the case k = 1. It was conjectured in [1] that this result remains true for all k > 0. We confirm this conjecture and prove that if Y ∼ Gamma(k) and X ∼ HM k are independent rvs, then Y • X ∼ GGC and Y /X ∼ GGC, for all k > 0 (Theorem 3.1).…”

On the Range of Exponential Functionals of Lévy Processes

“…Just recall Proposition 3.5 or the example mentioned in the introduction, which states that (0,∞) e −(σBt+at) dt has an inverse Gamma distribution which is a GGC, where (B t ) t≥0 is a Brownian motion and σ, a > 0. Further explicit examples of exponential functionals whose distributions are generalized Gamma convolutions can also be found in [7] and [4]. As generalized Gamma convolutions are selfdecomposable, one can also directly transfer the results from the last section to obtain conditions on GGCs to be in the range R ξ for a given process ξ.…”