Limit theorems for p-domain functionals of stationary Gaussian fields

Leonenko, Nikolai ORCID: https://orcid.org/0000-0003-1932-4091, Maini, Leonardo, Nourdin, Ivan and Pistolato, Francesca 2024. Limit theorems for p-domain functionals of stationary Gaussian fields. Electronic Journal of Probability 29 , 136. 10.1214/24-EJP1197

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Abstract

Fix an integer p ≥ 1 and refer to it as the number of growing domains. For each i ∈ { 1 , … , p } , fix a compact subset D i ⊆ R d i where d 1 , … , d p ≥ 1 . Let d = d 1 + ⋯ + d p be the total underlying dimension. Consider a continuous, stationary, centered Gaussian field B = ( B x ) x ∈ R d with unit variance. Finally, let φ : R → R be a measurable function such that E [ φ ( N ) 2 ] < ∞ for N ∼ N ( 0 , 1 ) . In this paper, we investigate central and non-central limit theorems as t 1 , … , t p → ∞ for functionals of the form Y ( t 1 , … , t p ) : = ∫ t 1 D 1 × ⋯ × t p D p φ ( B x ) d x . Firstly, we assume that the covariance function C of B is separable (that is, C = C 1 ⊗ … ⊗ C p with C i : R d i → R ), and thoroughly investigate under what condition Y ( t 1 , … , t p ) satisfies a central or non-central limit theorem when the same holds for ∫ t i D i φ ( B ( i ) x i ) d x i for at least one (resp. for all) i ∈ { 1 , … , p } , where B ( i ) stands for a stationary, centered, Gaussian field on R d i admitting C i for covariance function. When φ is an Hermite polynomial, we also provide a quantitative version of the previous result, which improves some bounds from [31]. Secondly, we extend our study beyond the separable case, examining what can be inferred when the covariance function is either in the Gneiting class or is additively separable.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Publisher:	Institute of Mathematical Statistics
Funders:	ARC Discovery Grant, LMS grant, FAPESP, Luxembourg National Research Fund
Date of First Compliant Deposit:	30 August 2024
Date of Acceptance:	28 August 2024
Last Modified:	01 Oct 2024 11:00
URI:	https://orca.cardiff.ac.uk/id/eprint/171675

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