Summary: Fix an integer \(p \geq 1\) and refer to it as the number of growing domains. For each \(i \in \{1, \ldots, p \}\), fix a compact subset \(D_i \subseteq \mathbb{R}^{d_i}\) where \(d_1, \ldots, d_p \geq 1\). Let \(d = d_1 + \cdots + d_p\) be the total underlying dimension. Consider a continuous, stationary, centered Gaussian field \(B = (B_x )_{x \in \mathbb{R}^d}\) with unit variance. Finally, let \(\varphi : \mathbb{R} \to \mathbb{R}\) be a measurable function such that \(\mathbb{E} [\varphi (N)^2] < \infty\) for \(N \sim N(0, 1)\).
In this paper, we investigate central and non-central limit theorems as \(t_1, \ldots, t_p \to \infty\) for functionals of the form \[ Y(t_1, \ldots, t_p) : = \int_{t_1 D_1 \times \cdots \times t_p D_p} \varphi (B_x) dx. \] Firstly, we assume that the covariance function \(C\) of \(B\) is separable (that is, \(C = C_1 \otimes \ldots \otimes C_p\) with \(C_i : \mathbb{R}^{d_i} \to \mathbb{R})\), and thoroughly investigate under what condition \(Y(t_1, \ldots, t_p)\) satisfies a central or non-central limit theorem when the same holds for \(\int_{t_i D_i} \varphi (B_{x_i}^{(i)}) dx_i\) for at least one (resp. for all) \(i \in \{1, \ldots, p\}\), where \(B^{(i)}\) stands for a stationary, centered, Gaussian field on \(\mathbb{R}^{d_i}\) admitting \(C_i\) for covariance function. When \(\varphi\) 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.