${\mathbb{R}}$ (${\mathbb{R}}_+$) is denoted by ${\mathcal{P}}$ (${\mathcal{P}}^+$).\
For a real-valued Lévy process $(\xi_t)_{t\geq 0}$, the [*characteristic exponent*]{} is given by its Lévy-Khintchine formula (e.g. [@sato Thm. 8.1]) $$\begin{aligned}
 \label{levykhintchine}
\log \phi_\xi(u) &:= \log {\mathbb{E}}\left[e^{i u \xi_1 } \right]\\
&= i \gamma_\xi u - \frac{1}{2} \sigma_\xi^2 u^2   + \int_{{\mathbb{R}}}
(e^{iux} -1 -i ux  \mathds{1}_{|x|\leq 1}) \nu_\xi(dx), \quad u\in {\mathbb{R}}, \nonumber\end{aligned}$$ where $(\gamma_\xi, \sigma_\xi^2, \nu_\xi)$ is the [*characteristic triplet*]{} of the Lévy process $\xi$. We refer to [@sato] for further information on Lévy processes.\
In the special case of a subordinator $(\eta_t)_{t\geq 0}$, i.e. of a nondecreasing Lévy process, we will also use its Laplace transform which we denote as ${\mathbb{L}}_{\eta} (u) := {\mathbb{L}}_{\eta_1} (u)={\mathbb{E}}[e^{-u\eta_1}]= e^{-\psi_\eta(u)}$, $u\geq 0$, where the [*Laplace exponent*]{} $\psi_\eta$ is a [*Bernstein function*]{} (BF), i.e. $$\label{eq:BF} 
 \psi_\eta(u)=a_\eta u + \int_{(0,\infty)} (1-e^{-ut})\nu_\eta(dt), \quad u>0,$$ with $a\geq 0$ called the [*drift*]{} of $\eta$ and a Lévy measure $\nu_\eta$. A thorough introduction to BFs can be found in the monograph [@rene-book]. Remark that general BFs as defined in [@rene-book] may have an additional constant term, while in this article we restrict on BFs which are Laplace exponents of a probability measure, that is which are zero in zero and hence are 