Central limit theorems and the geometry of polynomials
Abstract
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 $\delta = \min_{\zeta}\zeta1$ 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 realstable probability generating functions.
 Publication:

arXiv eprints
 Pub Date:
 August 2019
 arXiv:
 arXiv:1908.09020
 Bibcode:
 2019arXiv190809020M
 Keywords:

 Mathematics  Probability;
 Mathematics  Classical Analysis and ODEs;
 Mathematics  Combinatorics
 EPrint:
 44 pages. Typo in abstract fixed