Multivariate CLT follows from strong Rayleigh property

Abstract

Let $(X_1,\dots ,X_d)$ be random variables taking nonnegative integer values and let $f(z_1,\dots ,z_d)$ be the probability generating function. Suppose that $f$ is real stable; equivalently, suppose that the polarization of this probability distribution is strong Rayleigh. In specific examples, such as occupation counts of disjoint sets by a determinantal point process, it is known~\cite{soshnikov02} that the joint distribution must approach a multivariate Gaussian distribution. We show that this conclusion follows already from stability of $f$.