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
$$
Z\_N(x) := \frac{1}{\sqrt{2N\sigma^2}} \left(\sum\_{i=1}^N X\_i(x)^2-N\sigma^2 \right),\;\forall x \in \Omega.
$$



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> **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 $N \to \infty$ ?
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Central limit theorem for chi-squared random field on $\mathbb R^p$
$\newcommand\si\sigma\newcommand\Om\Omega$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 $(R(x\_i,x\_j)/(2\si^2))\_{i,j=1}^k$.




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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](https://link.springer.com/book/10.1007/978-3-642-54075-2#about).


