Corpus ID: 201657133

Central limit theorems and the geometry of polynomials

  title={Central limit theorems and the geometry of polynomials},
  author={M. Michelen and J. Sahasrabudhe},
  journal={arXiv: Probability},
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… Expand
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