+ \int_{(0,\infty)} (1-e^{-u t}) \frac{k(t)}{t} d t, \quad u\geq 0,$$ for some $a\geq 0$ and a completely monotone (CM) function $k:(0,\infty)\to [0,\infty)$.\
The class of probability distributions whose Laplace transform is of the form for some $a\geq 0$ with $\frac{k(t)}{t}$ CM is called Goldie-Steutel-Bondesson class or simply Bondesson’s class (${\mathrm{BO}}$). Its Laplace exponents are referred to as complete Bernstein functions (CBF) and they can always be represented as $$\label{eq:completebernstein}
 \psi_\mu(u) = a u +  \int_{(0,\infty)} \frac{u}{u + x} d \rho(x), \quad u\geq 0,$$ with $a\geq 0$ and a so-called Stieltjes measure $\rho$, that is a measure $\rho$ on $(0,\infty)$ for which $\int_{(0,\infty)} (1+x)^{-1} \rho(dx)<\infty$. For further details and an overview of the existing literature we refer to [@maejimalevymatters] and [@rene-book].\
Recall that ${\mathrm{BO}}$ is the smallest class of distributions which contains all mixtures of exponential distributions and is closed under convolutions and weak limits, while ${\mathrm{T}}$ is the smallest class that contains all gamma distributions and is closed under convolutions and weak limits. Also recall that ${\mathrm{T}}\subset {\mathrm{BO}}\subset {\mathrm{ID}}^+$ and ${\mathrm{T}}\subset {\mathrm{L}}^+ \subset {\mathrm{ID}}^+$, but ${\mathrm{L}}^+\not\subset {\mathrm{BO}}$ and ${\mathrm{BO}}\not\subset {\mathrm{L}}^+$.

Generalized Gamma convolutions and distributions in ${\mathrm{BO}}$ are connected via exponential functionals as shown in the following proposition, which has 