Suppose a function $G:\mathbb{R}^d\rightarrow\mathbb{R}$ is given.

What are some necessary or sufficient conditions on $G$ for there to exist a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a jointly-measurable function $V:\Omega\times\mathbb{R}^d\rightarrow \mathbb{R}$ that realizes a spatially homogenous Gaussian random field with $\mathbb{E}(V(\cdot,x)) = 0$ and $\mathbb{E}(V(\cdot,x)V(\cdot,y)) = G(x-y)$?

  • $\begingroup$ This is a special case of the Bochner-Minlos Theorem. $\endgroup$
    – user1504
    Oct 31 '20 at 2:07

The necessary and sufficient condition is for the function $G$ to be positive definite, that is, for the matrix $(G(x_j-x_k))_{j,k\in[n]}$ to be positive definite for any natural $n$ and any distinct $x_1,\dots,x_n$ in $\mathbb R^d$, where $[n]:=\{1,\dots,n\}$.


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