Multivariate CLT follows from strong Rayleigh property

TL;DR

This paper proves that the multivariate central limit theorem for certain discrete distributions can be derived solely from the stability property of their probability generating functions, linking stability to Gaussian convergence.

Contribution

It establishes that the stability of the probability generating function implies the multivariate CLT, extending known results from determinantal point processes to a broader class.

Findings

Multivariate CLT follows from stability of the generating function.

Stability implies Gaussian convergence for nonnegative integer-valued variables.

Links between stability and probabilistic limit theorems are established.

Abstract

Let $(X_1 , \ldots , X_d)$ be random variables taking nonnegative integer values and let $f(z_1, \ldots , z_d)$ 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 $f$.