Limit theorems for p-domain functionals of stationary Gaussian fields

Autor(es)

Leonenko, Nikolai ; Maini, Leonardo ; Nourdin, Ivan ; Pistolato, Francesca

• Tipo de documento: Artigo Científico
• Fonte: ELECTRONIC JOURNAL OF PROBABILITY; v. 29, p. 34-pg., 2024-01-01.

Resumo

Fix an integer p >= 1 and refer to it as the number of growing domains. For each i is an element of{1,& mldr;,p}, fix a compact subset D-i subset of R( di )where d(1),& mldr;,d(p)>= 1. Let d=d(1)+& ctdot;+d(p) be the total underlying dimension. Consider a continuous, stationary, centered Gaussian field B=(Bx)(x is an element of Rd) with unit variance. Finally, let phi:R -> R be a measurable function such that E[phi(N)(2)]<infinity for N similar to N(0,1). In this paper, we investigate central and non-central limit theorems as t(1),& mldr;,t(p)->infinity for functionals of the form Y(t1,& mldr;,tp):= integral(t1D1x & ctdot;xtpDp )phi(Bx)dx. Firstly, we assume that the covariance function C of B is separable (that is, C=C1 circle times & mldr;circle times Cp with Ci:R-di -> R), and thoroughly investigate under what condition Y(t1,& mldr;,tp)Y(t1,& mldr;,tp) satisfies a central or non-central limit theorem when the same holds for integral(tiDi)phi(B-xi((i)))dxi for at least one (resp. for all) i is an element of{1,& mldr;,p}, where B(i) stands for a stationary, centered, Gaussian field on Rdii admitting Ci for covariance function. When phi 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. (AU)

• Processo FAPESP: 22/09201-8 - Sistemas diferenciais fracionários e estocástica
• Beneficiário: Jayme Morandi Vaz
• Modalidade de apoio: Auxílio à Pesquisa - Pesquisador Visitante - Internacional