Let $\rho$ be zero-mean Gaussian random process with a squared exponential autocovariance function:
$$K_{\rho}(z_1, z_2) = e^{-\frac{(z_2-z_1)^2}{a^2}}$$
for some correlation length $a$.
Let $g(t,x)$ be a random field defined for $x\ge0$, $t\ge0$ and given in terms of the Gaussian random process $\rho$ and its first derivative.
$$g(t, x) = \frac{1}{x^2} \left[ 4 \pi \rho \left(\frac{1}{2}(ct + x)\right) + ct\dot{\rho}\left(\frac{1}{2}(ct + x)\right)\right]$$
where $c$ is a positive constant.
The goal is to find the autocovariance of the random field $g$ given by: $$K_g(t_1, x_1; t_2, x_2)=\mathbb{E}[g(t_1, x_1)g(t_2, x_2)]$$ I understand that the one can do so eventually after some tedious algebra/differentiation but my question is whether it can be shown to be positive definite? And perhaps my first question should be: how should (or if, one should) define positive definiteness for the autocovariance function of a multidimensional random process (i.e., a random field)?