# Sufficient conditions to be a covariance

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.

• 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? Dec 14, 2019 at 3:50
• 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.
– g g
Dec 14, 2019 at 12:00
• 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. Dec 14, 2019 at 14:51
• 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. Dec 14, 2019 at 16:30
• The footnote on page 6 here may be of interest: statweb.stanford.edu/~jtaylo/courses/stats352/notes/… Dec 18, 2019 at 16:41

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).
• I don't think it applies to the OP's function for all $p > 0$, wolframalpha.com/input/… Dec 14, 2019 at 19:25