Gaussian limit for determinantal point processes with $J$-Hermitian kernels

TL;DR

This paper proves a central limit theorem demonstrating that linear statistics of certain determinantal point processes with $J$-Hermitian kernels converge to a Gaussian distribution, under broad conditions.

Contribution

It establishes the Gaussian limit for linear statistics of $J$-Hermitian determinantal point processes on unions of Euclidean spaces, extending previous results to more general kernels.

Findings

Gaussian limit for linear statistics proven

Applicable to union of two Euclidean spaces

Valid for translation-invariant $J$-Hermitian kernels

Abstract

We show that the central limit theorem for linear statistics over determinantal point processes with $J$-Hermitian kernels holds under fairly general conditions. In particular, We establish Gaussian limit for linear statistics over determinantal point processes on union of two copies of $\mathbb{R}^d$ when the correlation kernels are $J$-Hermitian translation-invariant.