Central limit theorems from the roots of probability generating functions

For each $n$, let $X_n \in \{0,\ldots,n\}$ be a random variable with mean $\mu_n$, standard deviation $\sigma_n$, and let $$P_n(z) = \sum_{k=0}^n \mathbb{P}( X_n = k) z^k ,$$ be its probability generating function. We show that if none of the complex zeros of the polynomials $\{ P_n(z)\}$ are contained in a neighbourhood of $1 \in \mathbb{C}$ and $\sigma_n > n^{\varepsilon}$ for some $\varepsilon >0$, then $X_n^* =(X_n - \mu_n)\sigma^{-1}_n$ tends to a normal random variable $Z \sim \mathcal{N}(0,1)$ in distribution as $n \rightarrow \infty$. Moreover, we show this result is sharp in the sense that there exist sequences of random variables $\{X_n\}$ with $\sigma_n > C\log n$ for which $P_n(z)$ has no roots near $1$ and $X_n^*$ is not asymptotically normal. These results disprove a conjecture of Pemantle and improve upon various results in the literature. We go on to prove several other results connecting the location of the zeros of $P_n(z)$ and the distribution of the random variables $X_n$.