Central limit theorems and the geometry of polynomials

Let X \in \{0,\ldots,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}\leq t) - \mathbb{P}( Z \leq 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}\leq t) - \mathbb{P}( Z \leq 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.