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

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.