Limit theorems for $p$-domain functionals of stationary Gaussian fields

Abstract: 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 $d_1,\ldots,d_p\ge 1$. Let $d= d_1+\dots+d_{p}$ 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 $\varphi:\mathbb R \rightarrow \mathbb R$ be a measurable function such that $\mathrm E[\varphi(N)^2]<\infty$ 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,\dots,t_p):=\int{t_1D_1\times\dots \times t_pD_p}\varphi(B_{x})dx. ] Firstly, we assume that the covariance function $C$ of $B$ is {\it 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 $Y(t_1,\dots,t_p)$ satisfies a central or non-central limit theorem when the same holds for $\int_{t_iD_i}\varphi(B^{(i)}_{x_i})dx_i$ 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 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.