Anti-concentration of random variables from zero-free regions

TL;DR

This paper establishes a link between the zero distribution of a probability generating function and the anti-concentration properties of the associated random variable, providing new bounds on variance based on zero-free regions.

Contribution

It introduces a novel connection between the roots of the probability generating function and the variance, extending anti-concentration results beyond sums of i.i.d. variables.

Findings

Variance lower bound in terms of zero-free regions

Sharpness of the variance bound up to a constant factor

Extension of Littlewood–Offord type theorems to non-i.i.d. variables

Abstract

This paper provides a connection between the concentration of a random variable and the distribution of the roots of its probability generating function. Let $X$ be a random variable taking values in $\{0,\ldots,n\}$ with $\mathbb{P}(X = 0)\mathbb{P}(X = n) > 0$ and with probability generating function $f_X$. We show that if all of the zeros $\zeta$ of $f_X$ satisfy $|\arg(\zeta)| \geq \delta$ and $R^{-1} \leq |\zeta| \leq R$ then \[ \operatorname{Var}(X) \geq c R^{-2\pi/\delta}n, \] where $c > 0$ is a absolute constant. We show that this result is sharp, up to the factor $2$ in the exponent of $R$. As a consequence, we are able to deduce a Littlewood--Offord type theorem for random variables that are not necessarily sums of i.i.d.\ random variables.