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October 2021 Non-universal fluctuations of the empirical measure for isotropic stationary fields on S2×R
Domenico Marinucci, Maurizia Rossi, Anna Vidotto
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Ann. Appl. Probab. 31(5): 2311-2349 (October 2021). DOI: 10.1214/20-AAP1648

Abstract

In this paper, we consider isotropic and stationary real Gaussian random fields defined on S2×R and we investigate the asymptotic behavior, as T+, of the empirical measure (excursion area) in S2×[0,T] at any threshold, covering both cases when the field exhibits short and long memory, that is, integrable and nonintegrable temporal covariance. It turns out that the limiting distribution is not universal, depending both on the memory parameters and the threshold. In particular, in the long memory case a form of Berry’s cancellation phenomenon occurs at zero-level, inducing phase transitions for both variance rates and limiting laws.

Funding Statement

DM and AV acknowledge the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome “Tor Vergata”, CUP E83C18000100006. The research of MR has been supported by the INdAM-GNAMPA Project 2019 Proprietà analitiche e geometriche di campi aleatori and the ANR-17-CE40-0008 Project Unirandom.

Citation

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Domenico Marinucci. Maurizia Rossi. Anna Vidotto. "Non-universal fluctuations of the empirical measure for isotropic stationary fields on S2×R." Ann. Appl. Probab. 31 (5) 2311 - 2349, October 2021. https://doi.org/10.1214/20-AAP1648

Information

Received: 1 March 2020; Revised: 1 September 2020; Published: October 2021
First available in Project Euclid: 29 October 2021

Digital Object Identifier: 10.1214/20-AAP1648

Subjects:
Primary: 60G60
Secondary: 33C55 , 60D05 , 60F05

Keywords: Berry’s cancellation , central and noncentral limit theorems , empirical measure , Sphere-cross-time random fields

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 5 • October 2021
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