Central limit theorems from the roots of probability generating functions

Marcus Michelen, Julian Sahasrabudhe

For each $n$, let $X_n\in \{0,\dots ,n\}$ 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 prob…