3
$\begingroup$

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

$\endgroup$
1
  • $\begingroup$ This is a special case of the Bochner-Minlos Theorem. $\endgroup$
    – user1504
    Oct 31 '20 at 2:07
5
$\begingroup$

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 $[n]:=\{1,\dots,n\}$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.