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 $V$, the unnormalized autocorrelation function of a random field isIt depends only on the separation $h$, and $C(0)$ is the pointwise variance. If $C(0)>0$, the normalized correlation is $\rho (h)=C(h)/C(0)$; the two conventions have different units.

For a mean-square continuous field, its spectral measure of a stationary random field $\nu$ obeys $C(h)=\int e^{ik\cdot h}\nu (dk)$. If $C$ is integrable, an ordinary power spectral density exists and, in the angular-wavevector convention,Thus $\nu$ has density $S(k)/(2\pi {)}^d$. A spectral measure need not have a density: a spatially constant centered Gaussian field of variance $a>0$ has $C(h)=a$ and spectral measure $a{\delta }_0$, concentrated at zero wavevector. Calling every spectrum an ordinary function would miss this case.^[ref]

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.