Central limit theorem for chi-squared random field on $\mathbb R^p$ Let $X:x \mapsto X(x)$ be a centered stationary Gaussian process on the $\Omega:=\mathbb R^p$, such that $X(x) \overset{d}{=}X(x')$ for all $x,x' \in \Omega$. Set $\sigma^2 := \mbox{Var}(X(0)) = \mathbb E[X(0)^2]$. Let the random fields $X_1,\ldots,X_N$ be iid copies of $X$, and define a random process $Z_N$ on $\Omega$ by
$$
Z_N(x) := \frac{1}{\sqrt{2N\sigma^2}} \left(\sum_{i=1}^N X_i(x)^2-N\sigma^2 \right),\;\forall x \in \Omega.
$$

Question. Is there a central limit theorem (perhaps under further conditions on the base field $X$) for the limiting distribution of the random field $Z_N$ when $N \to \infty$ ?

 A: $\newcommand\si\sigma\newcommand\Om\Omega$To determine the limit distribution of the process $Z_N$ (and even the distribution of the process $X$), it is not enough to know only $\si^2$; one also has to know the covariances $r_{x,y}:=Cov(X(x),X(y))$ for $x,y$ in $\Om$.
Then, using e.g. the joint moment generating function of $(X(x),X(y))$, one can see that $$Cov(X(x)^2,X(y)^2)=R(x,y):=2r_{x,y}^2.$$
So, by the multivariate central limit theorem, for each $(x_1,\dots,x_k)\in\Om^k$, the corresponding finite-dimensional distribution of the process $Z_N$ (that is, the joint distribution of $(Z_N(x_1),\dots,Z_N(x_k))$) will converge to the corresponding finite-dimensional distribution of a centered Gaussian process $Z$ (that is, the joint distribution of $(Z(x_1),\dots,Z(x_k))$), which will be the $k$-variate normal distribution with zero mean and covariance matrix $(R(x_i,x_j)/(2\si^2))_{i,j=1}^k$.

To address such questions as into what functional vector space $V$ can the realizations of the limit process $Z$ and even of the original process $X$ be placed and whether one has the convergence of the full distributions of the processes $Z_N$ to the full distribution of the process $Z$ in $V$ require specific information on the covariances $r_{x,y}$. See e.g. Talagrand's book.
