Stability of Bernstein's Theorem and Soft Doubling for Vector Gaussian Channels

TL;DR

This paper extends Bernstein's stability theorem to vector Gaussian distributions using characteristic functions and introduces a soft doubling method to prove Gaussian optimality in certain noisy communication channels without needing capacity-achieving distributions.

Contribution

It generalizes Bernstein's stability to vectors and develops a novel soft doubling approach for Gaussian optimality in vector channels.

Findings

Extended stability results to vector Gaussian distributions.

Developed a soft doubling argument for Gaussian optimality.

Proved Gaussian vectors are optimal in specific noisy channels.

Abstract

The stability of Bernstein's characterization of Gaussian distributions is extended to vectors by utilizing characteristic functions. Stability is used to develop a soft doubling argument that establishes the optimality of Gaussian vectors for certain communications channels with additive Gaussian noise, including two-receiver broadcast channels. One novelty is that the argument does not require the existence of distributions that achieve capacity.