Sufficient Conditions for Existence of $J_{\alpha }(X+\sqrt[\alpha ]{\eta }N)$

In his technical report~\cite[sec. 6]{barrontech}, Barron states that the de Bruijn’s identity for Gaussian perturbations holds for any RV having a finite variance. In this report, we follow Barron’s steps as we prove the existence of $J_{\alpha }(X+\sqrt[\alpha ]{\eta }N)$, $\eta >0$ for any Radom Variable (RV) $X\in \mathcal{L}$ where \begin{equation*} $\mathcal{L}$ = \left{ $\text{RVs}$ ,,U: \int \ln\left(1 + |U|\right),dF_{U}(u) $\text{ is finite }$ \right}, \end{equation*} and where $N\sim \mathcal{S}(\alpha ;1)$ is independent of $X$, $0<\alpha <2$.