Local tail bounds for polynomials on the discrete cube

TL;DR

This paper proves a local tail bound for polynomials of Bernoulli variables on the discrete cube, confirming a conjecture and refining existing exponential decay bounds for their distribution tails.

Contribution

It establishes a local tail bound for degree-d polynomials of Bernoulli variables, confirming Keller and Klein's conjecture and improving understanding of their quantile behavior.

Findings

Proves a local tail bound for polynomials on the discrete cube.

Confirms a conjecture by Keller and Klein.

Refines the exponential decay rate of tail probabilities.

Abstract

Let $P$ be a polynomial of degree $d$ in independent Bernoulli random variables which has zero mean and unit variance. The Bonami hypercontractivity bound implies that the probability that $|P| > t$ decays exponentially in $t^{2/d}$. Confirming a conjecture of Keller and Klein, we prove a local version of this bound, providing an upper bound on the difference between the $e^{-r}$ and the $e^{-r-1}$ quantiles of $P$.