measure $\mu$) is [*c-decomposable*]{}, $c\in(0,1)$, or semi-selfdecomposable if holds for a $c\in(0,1)$ and a random variable $Y_c$ such that ${\mathcal{L}}(Y_c) \in {\mathrm{ID}}$. We write ${\mathrm{L}}(c), {\mathrm{L}}^+(c)$ for the class of $c$-decomposable distributions on ${\mathbb{R}}$ and ${\mathbb{R}}_+$, respectively. As in the case of selfdecomposable distributions, we refer to the random variable $Y_c$ in as the [*c-factor*]{} of $X$.

By [@sato Prop. 15.5] it holds ${\mathrm{L}}(c)\subset {\mathrm{ID}}$.

For probability distributions on ${\mathbb{R}}_+$ one can characterize $c$-decomposability in terms of the Laplace exponents. In particular, $\mu\in {\mathrm{L}}^+(c)$ if and only if $\psi_{\mu_c}(u)=\psi_\mu(u)-\psi_\mu(cu)$, $u>0$, is a BF. The fact that BFs build a convex cone then implies directly ${\mathrm{L}}^+(c) \subseteq {\mathrm{L}}^+_{c^n}$ for all $n\in{\mathbb{N}}$. More detailed $$\label{eq:cnfactor}
 \psi_{\mu_{c^n}}(u)=\sum_{i=0}^{n-1} \psi_{\mu_c} (c^i u)$$ is the Laplace exponent of the $c^n$-factor of $\mu\in {\mathrm{L}}^+(c)$. Using this one further obtains for any $\mu\in {\mathrm{L}}^+(c)$ $$\psi_\mu (u)= \lim_{n\to \infty} \psi_{\mu_{c^n}}(u) = \lim_{n\to\infty} \sum_{i=0}^{n-1} \psi_{\mu_c}(c^i u)$$ such that $${\mathrm{L}}^+(c)=\{ \mu \in {\mathcal{P}}^+, \mbox{ s.t. } \psi_\mu(u) =  \sum_{i=0}^{\infty} f(c^i u)  \mbox{ for some BF }f\}.$$

Generalized Gamma Convolutions
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The class of generalized Gamma convolutions ${\mathrm{T}}$ is a subclass of the selfdecomposable distributions in ${\mathcal{P}}^+$. In particular, every $\mu\in {\mathrm{T}}$, has a Laplace exponent of the form $$\label{eq:thorinbernstein}
 \psi_\mu(u)= au 