An elementary proof of anti-concentration for degree two non-negative Gaussian polynomials

TL;DR

This paper provides an elementary proof of the anti-concentration inequality for degree two non-negative Gaussian polynomials, simplifying understanding of their probabilistic behavior.

Contribution

It offers a simplified, elementary proof of a known anti-concentration result specifically for degree two non-negative Gaussian polynomials.

Findings

Proves anti-concentration for degree two non-negative Gaussian polynomials

Simplifies the proof of a classical anti-concentration inequality

Enhances understanding of Gaussian polynomial behavior

Abstract

A classic result by Carbery and Wright states that a polynomial of Gaussian random variables exhibits anti-concentration in the following sense: for any degree $d$ polynomial $f$, one has the estimate $P( |f(x)| \leq \varepsilon \cdot E|f(x)| ) \leq O(1) \cdot d \varepsilon^{1/d}$, where the probability is over $x$ drawn from an isotropic Gaussian distribution. In this note, we give an elementary proof of this result for the special case when $f$ is a degree two non-negative polynomial.