# 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

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

## Related Questions

### Examples of “non equivalent” algebras that are derived equivalent?

0  Asked on November 3, 2021 by marco-farinati

### Equivalence of families indexes of Fredholm operators

1  Asked on November 3, 2021 by rodrigo-dias

### An example of a special $1$-dimensional non-Noetherian valuation domain

1  Asked on November 3, 2021 by s-t-stanly

### What are some interesting applications of the Archimedean Property?

0  Asked on November 3, 2021 by fisura-filozofica

### Origins of the “baby Freiman” theorem

0  Asked on November 3, 2021 by seva

### Is there a (discrete) monoid M injecting into its group completion G for which BM is not homotopy equivalent to BG?

0  Asked on November 3, 2021 by omar-antoln-camarena

### Unrestricting The Parameters of a Functional Equation

0  Asked on November 3, 2021

### Pseudoreflection groups in affine varieties

0  Asked on November 3, 2021

### Image of function contains identity elements

0  Asked on November 3, 2021 by pi66

### Integrate Radon-Nikodým derivatives against Lebesgue measure

1  Asked on November 3, 2021

### Higher derivatives of weighted zeta function and continuation to $s=1$

0  Asked on November 3, 2021 by milo-moses

### Distance properties of the permutations of a set of points in a Euclidean space

0  Asked on November 3, 2021

### Roots of determinant of matrix with polynomial entries — a generalization

0  Asked on November 3, 2021

### Laplacian coupled with another equation over a two-dimensional rectangular region

0  Asked on November 3, 2021

### Is there a finite equational basis for the join of the commutative and associative equations?

0  Asked on November 3, 2021

### What are some examples of proving that a thing exists by proving that the set of such things has positive measure?

9  Asked on November 3, 2021 by tom-leinster

### Literature and history for: lifting matrix units modulo various kinds of ideal

0  Asked on November 3, 2021 by yemon-choi

### Smallest family of subsets that generates the discrete topology

2  Asked on November 3, 2021 by user175348

### A global mathematics library

3  Asked on November 3, 2021