# Combinatorics and Probability Seminar: Multivariate Limit Theorems and Large Variance from Zero Free Regions, by Marcus Michelen

September 16, 2019

3:00 PM - 3:50 PM

Marcus Michelen (UIC): Multivariate Limit Theorems and Large Variance from Zero Free Regions

We prove a multivariate central limit theorem for random variables $(X_1,\ldots,X_d)$ that are \emph{strong Rayleigh}, a higher-dimensional analogue of random variables with real-rooted generating function. The strong Rayleigh property can be thought of as a notion of negative dependence and this limit theorem was conjectured by Ghosh, Liggett and Pemantle. I will present a short history of the strong Rayleigh property and demonstrate how this multivariate limit theorem can be reduced to a univariate limit theorem discussed in the previous week.

Back in the univariate setting, let $X \in \{0,\ldots,n\}$ be a random variable and $f_X$ be its probability generating function. We show that if the roots of $f_X$ have argument bounded away from $0$, then the variance of

$X$ must be remarkably large. This takes the form of a lower bound that is sharp up to constants, and is proved using similar tools used to prove the limit theorems of the previous week. This talk is based on joint work with Julian Sahasrabudhe.

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## Date posted

Sep 20, 2019

## Date updated

Sep 20, 2019