Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sum of Gaussian vectors and large sets

Antoine Song

Abstract

We show that for some constant $κ>0$, any centered $κ$-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 $Λ\geq 1$, any centered random vector $X$ in $\mathbb{R}^n$ such that $\|X\|\leq Λ$ almost surely and $\|\mathrm{Cov}(X)\|\leq {Λ^2 }{e^{-Λ^2}}$ 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.

Sum of Gaussian vectors and large sets

Abstract

We show that for some constant , any centered -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 , any centered random vector in such that almost surely and 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.
Paper Structure (18 sections, 26 theorems, 131 equations)

This paper contains 18 sections, 26 theorems, 131 equations.

Table of Contents

  1. Conventions and notations:
  2. Equivalence
  3. How expressive are sums of Gaussian vectors?
  4. Basic properties of sums of Gaussian vectors
  5. Real-valued subgaussian random variables
  6. Random vectors with bounded norm and covariance
  7. Further discussions
  8. Geometric interpretation
  9. Averages of Gaussians
  10. Simple random vectors
  11. How large are sums of large sets?
  12. Large permutation invariant sets
  13. Ellipsoids in sums of large sets
  14. Further discussions
  15. Largest slices
  16. ...and 3 more sections

Key Result

Theorem 3

There is a universal constant $\kappa>0$ such that given any centered real-valued $\kappa$-subgaussian random variable $X$, there are three standard Gaussian random variables $G_1,G_2,G_3$ with

Theorems & Definitions (46)

  • Theorem 3: Three Gaussians
  • Theorem 4: Norm vs Covariance
  • Corollary 5: Permutation invariant sets
  • Corollary 6: Largest Ellipsoid
  • Theorem 1.1: Equivalence
  • Lemma 1.2: Steinhaus Lemma
  • proof
  • Lemma 1.3: Volume of neighborhood
  • Theorem 1.4: Tal21
  • Theorem 1.5: Dad19
  • ...and 36 more