Sufficient Conditions for Existence of $J_{\hat{I}}\pm (X+\sqrt[\hat{I}\pm ]{\hat{I}}\cdot N)$

Abstract

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 $J_{\hat{I}}\pm \left(X+\sqrt[\hat{I}\pm ]{\hat{I}}\cdot N\right)$, $\hat{I}\cdot >0$ 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 $N\sim \mathcal{S}(\hat{I}\pm ;1)$ is independent of $X$, $0<\hat{I}\pm <2$.