Regularized Distributions and Entropic Stability of Cramer's Characterization of the Normal Law

TL;DR

This paper proves the stability of Cramer's characterization of the normal distribution using regularized distributions, focusing on total variation and entropic distances, and refines related theorems for variables with finite second moments.

Contribution

It introduces stability results for the normal law characterization under regularized distributions and improves Sapogov-type theorems for finite second moment variables.

Findings

Stability of normal law characterization in total variation norm.

Stability of normal law characterization in entropic distance.

Refined Sapogov-type theorems for finite second moment variables.

Abstract

For regularized distributions we establish stability of the characterization of the normal law in Cramer's theorem with respect to the total variation norm and the entropic distance. As part of the argument, Sapogov-type theorems are refined for random variables with finite second moment.