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

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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.

### Author and article information

###### Journal
14 January 2023
###### Article
2301.05992
f26a2364-6c53-44ab-8fbf-2cb02726eb29