# Existence of Gaussian random field with prescribed covariance

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 $$\mathbb{E}(V(\cdot,x)V(\cdot,y)) = G(x-y)$$?

• This is a special case of the Bochner-Minlos Theorem. 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 $$[n]:=\{1,\dots,n\}$$.