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.



Sufficient conditions to be a covariance
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). 


