Central limit theorems and the geometry of polynomials

Abstract

Let $X\in \{0,\dots ,n\}$ be a random variable with mean $\mu$, standard deviation $\sigma$, and let $f_X(z)={\sum }_k\mathbb{P}(X=k)z^k$ be its probability generating function. Pemantle conjectured that if $\sigma$ is large and $f_X$ has no roots close to $1\in \mathbb{C}$, then $X$ must be approximately normal. We completely resolve this conjecture in the following strong quantitative form, obtaining sharp bounds. If $\delta ={\min }_{\zeta }|\zeta -1|$ over the complex roots $\zeta$ of $f_X$, and $X^{\ast }:=(X-\mu )/\sigma$, then ${\sup }_{t\in \mathbb{R}}|\mathbb{P}(X^{\ast }\le t)-\mathbb{P}(Z\le t)|=O((\log n)/(\delta \sigma ))$, where $Z\sim \mathcal{N}(0,1)$ is a standard normal variable. This gives the best possible version of a result of Lebowitz, Pittel, Ruelle and Speer. We also show that if $f_X$ has no roots with small argument, then $X$ must be approximately normal, again in a sharp quantitative form: if we set $\delta ={\min }_{\zeta }|\arg (\zeta )|$, then ${\sup }_{t\in \mathbb{R}}|\mathbb{P}(X^{\ast }\le t)-\mathbb{P}(Z\le t)|=O(1/(\delta \sigma ))$. Using this result, we answer a question of Ghosh, Liggett and Pemantle by proving a sharp multivariate central limit theorem for random variables with real-stable probability generating functions.