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 AnswersThe 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

0 Asked on November 3, 2021 by marco-farinati

1 Asked on November 3, 2021 by rodrigo-dias

fa functional analysis fredholm operators index theory kt k theory and homology vector bundles

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

ac commutative algebra ag algebraic geometry dedekind domains nonnoetherian valuation rings

0 Asked on November 3, 2021 by fisura-filozofica

0 Asked on November 3, 2021 by seva

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

at algebraic topology classifying spaces homotopy theory semigroups and monoids

0 Asked on November 3, 2021

analytic number theory co combinatorics generating functions

0 Asked on November 3, 2021

ag algebraic geometry geometric invariant theory orbifolds reference request reflection groups

1 Asked on November 3, 2021

limits and convergence measure theory stochastic calculus stochastic processes

0 Asked on November 3, 2021 by milo-moses

analytic continuation analytic number theory nt number theory

0 Asked on November 3, 2021

co combinatorics combinatorial optimization discrete geometry mg metric geometry permutations

0 Asked on November 3, 2021

0 Asked on November 3, 2021

differential operators elliptic pde heat equation linear pde mp mathematical physics

0 Asked on November 3, 2021

3 Asked on November 3, 2021 by tanya-vladi

fourier analysis fourier transform pr probability real analysis

9 Asked on November 3, 2021 by tom-leinster

0 Asked on November 3, 2021 by yemon-choi

2 Asked on November 3, 2021 by user175348

Get help from others!

Recent Answers

- Philipp on How do i draw a ray in unity
- kjetil b halvorsen on How to test consistency of responses?
- eric_kernfeld on How to test consistency of responses?
- DMGregory on MouseLook Script “Pops” back to the last value when the script is enabled after being disabled or destroyed
- Justin Markwell on Unity app crashes when using unmodified custom Android manifest (didn’t find class “UnityPlayerActivity”)

Recent Questions

- MouseLook Script “Pops” back to the last value when the script is enabled after being disabled or destroyed
- Unity app crashes when using unmodified custom Android manifest (didn’t find class “UnityPlayerActivity”)
- How do i draw a ray in unity
- How to test consistency of responses?
- How can I understand these variograms?

© 2022 AnswerBun.com. All rights reserved.