Limit theorems for p-domain functionals of stationary Gaussian fields

Fix an integer p 1 and refer to it as the number of growing domains. For each i\1, , p\, fix a compact subset Dᵢ R^dᵢ where d₁, , dₚ 1. Let d= d₁++d₏ be the total underlying dimension. Consider a continuous, stationary, centered Gaussian field B= (Bₓ) ₗ ₑ㵧 with unit variance. Finally, let: R R be a measurable function such that E (N) ²< for N N (0, 1). In this paper, we investigate central and non-central limit theorems as t₁, , tₚ for functionals of the form \ Y (t₁, , tₚ): =ₓ䃑₃䃑 ₓ䂹₃䂹 (Bₗ) dx. \ Firstly, we assume that the covariance function C of B is separable (that is, C=C₁ C₏ with Cᵢ: R^dᵢ R), and thoroughly investigate under what condition Y (t₁, , tₚ) satisfies a central or non-central limit theorem when the same holds for ₓ㶁₃㶁 (B^ (i) ₗ㶁) dxᵢ for at least one (resp. for all) i \1, , p\, where B^ (i) stands for a stationary, centered, Gaussian field on R^dᵢ admitting Cᵢ for covariance function. When is an Hermite polynomial, we also provide a quantitative version of the previous result, which improves some bounds from A. Reveillac, M. Stauch, and C. A. Tudor, Hermite variations of the fractional brownian sheet, Stochastics and Dynamics 12 (2012). 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.