Central limit theorems and the geometry of polynomials

Let $X \in \{0,\ldots,n \}$ 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 $$\sup_{t \in \mathbb{R}} \left|\mathbb{P}(X^{\ast} \leq t) - \mathbb{P}( Z \leq t) \, \right| = O\left(\frac{\log n}{\delta\sigma} \right)$$ 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 $$\sup_{t \in \mathbb{R}} \left|\mathbb{P}(X^{\ast} \leq t) - \mathbb{P}( Z \leq t) \, \right| = O\left(\frac{1}{\delta\sigma} \right).$$ 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.