\quad  \text{Thorin's class, generalized gamma convolutions (on } {\mathbb{R}}_+)\end{aligned}$$

Selfdecomposability
-------------------

A random variable $X$ (or equivalently a probability measure $\mu$) is called [*selfdecomposable*]{}, if for all $c\in(0,1)$, there exists a random variable $Y_c$, independent of $X$, such that $$\label{eq-def-SD}
X\overset{d}= cX'+Y_c,$$ where $X'$ is an independent copy of $X$. In this case we write $\mu={\mathcal{L}}(X)\in {\mathrm{L}}$. Obviously, for distributions on the positive real line, is equivalent to $${\mathbb{L}}_\mu(u)={\mathbb{L}}_\mu(cu){\mathbb{L}}_{\mu_c}(u), \quad u\geq 0, c\in (0,1),$$ or $$\label{eq-SD-bernsteindifferenz}
    \psi_\mu(u)-\psi_\mu(cu)=\psi_{\mu_c}(u), \quad u\geq 0, c\in (0,1),$$ where $\mu_c={\mathcal{L}}(Y_c)$. In particular it is known (cf. [@rene-book Prop. 5.17]), that every $\mu\in {\mathrm{L}}^+$ has a Laplace exponent of the form $$\label{eq:selfdecomposablebernstein}
 \psi_\mu(u)= au + \int_0^\infty (1-e^{-u t}) \frac{k(t)}{t} dt, \quad u\geq 0,$$ with $a\geq 0$ called the [*drift*]{} of $\mu$ and $k:[0,\infty)\to [0,\infty)$ non-increasing.\
The following proposition collects characterizations of selfdecomposable distributions in ${\mathcal{P}}^+$ which we intend to use in this paper. Most of them are well known. We couldn’t find characterization (iv) in this form in the literature, so we give a short instructive proof. Alternatively (iv) is easily seen to be equivalent to the characterization of selfdecomposability in [@steutelvanharn Thm. V.2.9]. Further characterizations of selfdecomposable distributions can also be found in [@maejimalevymatters; @satolevymatter; @steutelvanharn] 