Sufficient Conditions for Existence of $J_α(X + \sqrt[α]ηN)$

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 $J_α\left(X + \sqrt[α]ηN\right)$, $η> 0$ 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 $N \sim \mathcal{S}(α;1)$ is independent of $X$, $0< α<2$.

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