Central limit theorems and the geometry of polynomials

Abstract

Let $X\in \{0,\dots ,n\}$ be a random variable, with mean $\mu$ and 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 \[ \sup_{t \in \mathbb{R}} \left|\mathbb{P}(X^{\ast} \leq t) - \mathbb{P}( Z \leq t) \, \right| = O\left(\frac{\log n}{δσ} \right) \] where $Z\sim \mathcal{N}(0,1)$ is a standard normal. 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 \[ \sup_{t \in \mathbb{R}} \left|\mathbb{P}(X^{\ast} \leq t) - \mathbb{P}( Z \leq t) \, \right| = O\left(\frac{1}{δσ} \right). \] 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.

44 pages. Typo in abstract fixed