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Given a function, $c(x,y):\mathbb{R}\times \mathbb{R}\to \mathbb{R}$, what are sufficient conditions for this to be the covariance of some (centered) Gaussian random field $X:\mathbb{R}\to \mathbb{R}$, $$ c(x,y) = \mathbb{E}[X(x)X(y)] $$ Obviously, we would like $c$ to be symmetric, $c(x,y) =c(y,x)$, but what is also needed is that it be positive definite. I know there is Bochner's theorem, which relates $c$ to the Fourier transform of a positive measure, but my real question is, for closed form choices of $c$, are there simple conditions that can be easily checked (i.e. $c$ decays sufficiently rapidly as $|x-y|\to \infty$)?

EDIT: What I am really asking is, if I write down a simple function, like $$ c(x,y) = \frac{1}{1+|x-y|^p}, \quad p>0 $$ is there a way to tell, by inspection, whether or not there exists a Gaussian random field with this as its covariance kernel. I mention positive definiteness because, in my mind, it is not obvious how to check that $$ (Cf)(x) = \int c(x,y) f(y)dy,\quad f\in L^2(\mathbb{R}) $$ is a positive definite operator.

ALSO, I am not strictly wedded to working with $\mathbb{R}$. If there is a clear answer to this problem when $c:[0,1]^2\to\mathbb{R}$, I would be interested to hear that too.

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  • $\begingroup$ You say $X$ is a "random variable" but then you write $X(x)$. Do you mean that $X$ is a stochastic process indexed by $\mathbb{R}$? A stochastic process indexed by an uncountable set is a nasty object without further assumptions (e.g. cadlag); what do you want to impose here? $\endgroup$ Dec 14 '19 at 3:50
  • $\begingroup$ It is not clear what question you are asking. Is it what conditions on $c$ are required to make it a covariance of some random process (as the title suggests) or are you interested in criteria for positive definite functions (as the last paragraph suggests) ? Maybe you can rephrase your question to make this clear. $\endgroup$
    – g g
    Dec 14 '19 at 12:00
  • $\begingroup$ I tried to clarify this. 1. I am really thinking of Gaussian random fields. 2. In my mind, it is the positive definiteness of a given kernel that is the tough part to check. Perhaps that is incorrect. $\endgroup$ Dec 14 '19 at 14:51
  • $\begingroup$ In concrete examples, it is easy to disprove that a kernel of the form $c(x, y) = f(|x-y|)$ is positive definite, by computing the Fourier transform of $f$ up to some accuracy and looking for negative values. $\endgroup$ Dec 14 '19 at 16:30
  • $\begingroup$ The footnote on page 6 here may be of interest: statweb.stanford.edu/~jtaylo/courses/stats352/notes/… $\endgroup$ Dec 18 '19 at 16:41
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Polya's criterion says that if $f:\mathbb{R}\to \mathbb{R}$ is even, convex on $[0,\infty)$, with $f(0)>0$ and zero limit at infinity, then $c(x,y) = f(\vert x-y\vert)$ is a positive definite kernel, hence the existence of the Gaussian random field. It would apply to your function for $p$ less than $1$ for example (to be checked).

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  • $\begingroup$ I don't think it applies to the OP's function for all $p > 0$, wolframalpha.com/input/… $\endgroup$ Dec 14 '19 at 19:25
  • $\begingroup$ yes right, added a correction $\endgroup$
    – alesia
    Dec 14 '19 at 20:07
  • $\begingroup$ The kernel I wrote down was just for example, but this is exactly the sort of test I was looking for. $\endgroup$ Dec 16 '19 at 0:35

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