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

TL;DR

This paper establishes conditions under which the fractional Fisher information of a sum involving a random variable and a symmetric alpha-stable noise exists, extending de Bruijn's identity beyond finite variance cases.

Contribution

It proves the existence of fractional Fisher information for sums of arbitrary RVs with certain integrability conditions and alpha-stable noise, generalizing previous results.

Findings

Existence of $J_{\alpha}(X + \sqrt[\alpha]{\eta}N)$ under new conditions

Extension of de Bruijn's identity to broader class of random variables

Provides mathematical foundation for fractional Fisher information in stable noise models

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 $J_{\alpha}\left(X + \sqrt[\alpha]{\eta}N\right)$, $\eta > 0$ 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 $X$, $0< \alpha <2$.