Central limit theorems and the geometry of polynomials

Understand

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.