February 2024 Normality of smooth statistics for planar determinantal point processes
Antti Haimi, José Luis Romero
Author Affiliations +
Bernoulli 30(1): 666-682 (February 2024). DOI: 10.3150/23-BEJ1612

Abstract

We consider smooth linear statistics of determinantal point processes on the complex plane, and their large scale asymptotics. We prove asymptotic normality in the finite variance case, where Soshnikov’s theorem is not applicable. The setting is similar to that of Rider and Virág [Electron. J. Probab., 12, no. 45, 1238–1257, (2007)] for the complex plane, but replaces analyticity conditions by the assumption that the correlation kernel is reproducing. Our proof is a streamlined version of that of Ameur, Hedenmalm and Makarov [Duke Math J., 159, 31–81, (2011)] for eigenvalues of normal random matrices. In our case, the reproducing property is brought to bear to compensate for the lack of analyticity and radial symmetries.

Funding Statement

A. H. and J. L. R. gratefully acknowledge support from the Austrian Science Fund (FWF): Y 1199.

Citation

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Antti Haimi. José Luis Romero. "Normality of smooth statistics for planar determinantal point processes." Bernoulli 30 (1) 666 - 682, February 2024. https://doi.org/10.3150/23-BEJ1612

Information

Received: 1 January 2023; Published: February 2024
First available in Project Euclid: 8 November 2023

MathSciNet: MR4665593
zbMATH: 07788899
Digital Object Identifier: 10.3150/23-BEJ1612

Keywords: asymptotic normality , determinantal point process , linear statistics , Weyl-Heisenberg DPP

Vol.30 • No. 1 • February 2024
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