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Existence of Gaussian random field with prescribed covariance

Suppose a function $G:mathbb{R}^drightarrowmathbb{R}$ 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 $V:Omegatimesmathbb{R}^drightarrow mathbb{R}$ that realizes a spatially homogenous Gaussian random field with $mathbb{E}(V(cdot,x)) = 0$ and $mathbb{E}(V(cdot,x)V(cdot,y)) = G(x-y)$?

MathOverflow Asked by Cabbage on January 5, 2021

1 Answers

One Answer

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

Correct answer by Iosif Pinelis on January 5, 2021

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