Stationary Gaussian random field

A stationary Gaussian random field has Gaussian finite-dimensional distributions invariant under common spatial translations. Its mean is constant and its two-point covariance depends only on the separation of the points.

For a zero-mean stationary Gaussian random field $V(\mathbf{r})$, the autocorrelation function iswhich is independent of $\mathbf{r}$. Its Fourier transform is the spatial power spectral density.

For $W=n^2-⟨n^2⟩$, the covariance function is $⟨n^2{n}^{\prime 2}⟩-⟨n^2⟩⟨n^{\prime 2}⟩$. It involves fourth moments and is not determined by the two-point correlation of $n$ in general. For a centered Gaussian pair the fourth-moment identity reduces it to $2⟨n{n}^{\prime }{⟩}^2$. This extra Gaussian hypothesis must be stated before using that reduction.