Combinatorics and Probability Seminar
Marcus Michelen
UIC
Central Limit Theorems and the Geometry of Polynomials
Abstract: Let $X \in \{0,\ldots,n \}$ be a random variable and let
$$f_X(z) = \sum_{k} \mathbb{P}(X = k) z^k, $$
be its probability generating function. Pemantle conjectured that if the variance of $X$ is large and $f_X$ has no roots close to $1\in \mathbb{C}$ then
$X$ must be approximately normal. We will discuss a complete resolution of this conjecture in a strong quantitative form, thereby giving the best possible version of a result of Lebowitz, Pittel, Ruelle and Speer.
Additionally, if $f_X$ has no roots with small argument, then $X$ must be approximately normal, again in a sharp quantitative form. This talk is based on joint work with Julian Sahasrabudhe.
Monday September 9, 2019 at 3:00 PM in 612 SEO