Sum of Gaussian vectors and large sets

TL;DR

This paper proves that certain subgaussian variables can be expressed as sums of Gaussian variables, confirming a conjecture, and applies these results to convexity problems and geometric estimates in Gaussian spaces.

Contribution

It establishes that subgaussian variables are sums of Gaussian variables and solves related conjectures and geometric problems.

Findings

Centered subgaussian variables are sums of three Gaussian variables.

Centered vectors with bounded norm and covariance can be expressed as sums of Gaussian vectors.

Results settle a conjecture and provide optimal geometric estimates in Gaussian spaces.

Abstract

We show that for some constant $\kappa>0$, any centered $\kappa$-subgaussian random variable is equal to the sum of three standard Gaussian random variables, confirming a conjecture of M. Talagrand. We also prove that given $\Lambda\geq 1$, any centered random vector $X$ in $\mathbb{R}^n$ such that $\|X\|\leq \Lambda$ almost surely and $\|\mathrm{Cov}(X)\|\leq {\Lambda^2 }{e^{-\Lambda^2}}$ is equal to the sum of a universal number of standard Gaussian random vectors. In particular, a centered random vector is subgaussian if and only if it is a finite sum of Gaussian random vectors. We apply these results to settle the permutation invariant case of M. Talagrand's convexity problem, and to give optimal estimates on the largest ellipsoid contained in a sum of large sets in Gaussian spaces.