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.