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arXiv · 2301.05992

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

Stephen Tu·Ross Boczar
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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.

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Stephen Tu, Ross Boczar. 2023-01-15. An elementary proof of anti-concentration for degree two non-negative Gaussian polynomials. https://arxiv.org/abs/2301.05992

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