Stability of the Ghurye-Olkin Characterization of Vector Gaussian Distributions

TL;DR

This paper investigates how close to Gaussian the sum of vectors remains under approximate conditions of the Ghurye-Olkin characterization, with implications for entropy and source separation.

Contribution

It establishes stability results for the Ghurye-Olkin characterization of Gaussian vectors, extending classical theorems to near-Gaussian scenarios.

Findings

Sum of vectors is near-Gaussian if GO condition is approximately met

Any vector projection is near-Gaussian in distribution function domain

Results apply to stability of differential entropies and blind source separation

Abstract

The stability of the Ghurye-Olkin (GO) characterization of Gaussian vectors is analyzed using a partition of the vectors into equivalence classes defined by their matrix factors. The sum of the vectors in each class is near-Gaussian in the characteristic function (c.f.) domain if the GO independence condition is approximately met in the c.f. domain. All vectors have the property that any vector projection is near-Gaussian in the distribution function (d.f.) domain. The proofs of these c.f. and d.f. stabilities use tools that establish the stabilities of theorems by Kac-Bernstein and Cram\'er, respectively. The results are used to prove stability theorems for differential entropies of Gaussian vectors and blind source separation of non-Gaussian sources.