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

2023/01/15 by Stephen Tu, Tu, Stephen, Ross Boczar +1
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Probability (math.PR) #Statistical Methods and Inference #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2301.05992

openalex publication_date 2023/01/15 · openalex created_date 2023/02/15 · openalex updated_date 2026/07/28

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)| ≤ ε ⋅ E|f(x)| ) ≤ O(1) ⋅ d ε^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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