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

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.