Central limit theorems from the roots of probability generating functions

@article{Michelen2018CentralLT,
  title={Central limit theorems from the roots of probability generating functions},
  author={M. Michelen and J. Sahasrabudhe},
  journal={arXiv: Probability},
  year={2018}
}
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… Expand
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