Combinatorics and Probability Seminar
Marcus Michelen
UIC
Multivariate Limit Theorems and Large Variance from Zero Free Regions
Abstract: 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.
Monday September 16, 2019 at 3:00 PM in 612 SEO