this implies that $$\label{eq-SD-integral}
 \mu={\mathcal{L}}\left( \int_{(0,\infty)} e^{-t} dX_t \right).$$ Since $\mu$ exists by assumption and therefore the integral has to converge, we obtain ${\mathbb{E}}[\log^+(X_1)]<\infty$ and hence (v).\
Finally, $ \int_{(0,\infty)} e^{-t} dX_t$ is well known and easily seen to be selfdecomposable (see e.g. [@bertoinlindnermaller08]) which concludes the proof.

As already observed in [@BLMranges], Equation implies in particular, that $\mu$ and ${\mathcal{L}}(X)$ have the same drift and that the Lévy density of $\mu$ and the Lévy measure of $X$ are related by $$\label{eq:relationlevymeasures}
 k(t)=\nu_X((t,\infty))$$ (see also [@BarndorffNielsen-Shephard Eq. 4.17]).

\[factors\] [Differences of BFs as in (ii) and (iii) of the above proposition will appear frequently in the remaining sections of this article. Hence in the following, we refer to the distributions with Laplace exponent $\psi_{\mu_c}$ ($c\in(0,1)$) or $-\psi_{\mu_c}$ ($c>1$) as $c$-[*factor distributions*]{} of the distribution $\mu\in {\mathrm{L}}$. Recall that these are always in ${\mathrm{ID}}$ and that they are uniquely determined since $\mu \in {\mathrm{ID}}$.\
In terms of random variables we refer to $Y_c$ as the $c$-[*factor*]{} ($c\in(0,1)$) of $X$ if $X\overset{d}=cX'+Y_c$ and we say that $Y_c$ is the $c$-[*factor*]{} ($c>1$) for $X$, if $cX\overset{d}=X'+Y_c.$]{}

Further, from Theorem \[thm:density\] above we obtain the following characterization of densities of distributions in ${\mathrm{L}}_+$. The 