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A Random field is a family of random variables indexed by points of a spatial domain, or more generally by a multi-dimensional parameter set. When second moments are finite, its componentwise covariance function and spectral tensor describe second-order dependence. A Gaussian random field has jointly Gaussian finite-dimensional distributions; a general Random field need not be Gaussian.

A random phase screen models a thin random medium by multiplying a wave field by $e^{i\varphi (x)}$, where $\varphi$ is a real random field on the transverse plane. At the screen itself the modulus is unchanged, but later wave propagation converts phase variation into interference and amplitude variation.

If $\varphi$ is a zero-mean Gaussian random field, its phase factor has correlationThis follows from the characteristic function. The approximation neglects the medium's longitudinal thickness; its covariance and propagation model must be specified before predicting scattering or intensity statistics.

A sharp-k filter includes every Fourier mode below a cutoff with unit weight and excludes all modes above it. For a Gaussian random field, increasing the cutoff adds independent Fourier shells, so the smoothed density contrast is a Brownian motion when parameterized by its variance. The real-space window is not a localized spherical top-hat, so its halo mass assignment needs a specified convention.

For one selected polarization of waves, write the detected complex fields as $E_1(t),E_2(t+\tau )$ and their light intensities as $I_j=|E_j{|}^2$, absorbing a common conversion constant. With stationary ensemble averages and nonzero ${\bar{I}}_j=⟨I_j⟩$, defineHere second-order optical coherence concerns distinct detector channels, with the corresponding normal ordering in the quantum formulation. Independent detection has $g^{(2)}=1$. Subtracting the means gives the normalized covarianceThus multiplying raw photocurrents contains a large mean-product contribution; multiplying their fluctuations isolates the small excess. Independent detector shot noise does not supply a mean cross-channel excess, although it increases measurement uncertainty.

The displayed Siegert relation requires a zero-mean, jointly circular complex Gaussian random field, the usual chaotic thermal light model. Circular means invariance under a common arbitrary complex phase rotation; in particular $⟨E_i{E}_j⟩=0$. Apply the Gaussian moment pairing theorem to the real and imaginary parts:The remaining pairing vanishes by circularity. Division proves the Siegert relation. Gaussianity is an assumption about fourth moments, not a consequence of merely measuring first-order optical coherence; a coherent state or photon antibunching cannot be interpreted by this thermal formula.

For equal-intensity beams combined in a common optical mode with matched wave polarizations, interferometric visibility is $V=|g_{12}^{(1)}|$. Consequently ideal thermal stellar intensity interferometry measures $g^{(2)}-1=V^2$, while the full $g^{(2)}$ includes the independent-count reference one. Unpolarized thermal light with two independent equally populated wave polarizations has half this excess: adding their covariances doubles the numerator while multiplying total means quadruples the denominator. Finite detector response further suppresses the peak and must be calibrated.^[ref]

If the displayed covariance is finite and the mean obeys $m^{\prime }(x)+A(x)=0$, initialize the Musiela forward-curve equation with the corresponding Gaussian random field, independently of future noise. The resulting curve is stationary in time. Indeed shifting the initial covariance contributes the integral over $[t,\infty )$ and fresh noise contributes the integral over $[0,t]$, so their sum is $K$. A deterministic initial curve generally is not stationary, and nondecaying volatility may make $K(x,x)$ infinite. Exponentially decaying finite-factor volatilities give stationary Ornstein-Uhlenbeck processes.

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.

Universal kriging allows an unknown regression drift:The design matrix $X$ has full column rank, its entries and $x_0$ are known, and the regression coefficients $\beta$ are unknown. For example, rows $(1,s_i)$ and target row $(1,s_0)$ allow a linear spatial trend. Requiring $w^{T}Z$ to have zero bias for every $\beta$ gives $X^{T}w=x_0$. With known positive-definite matrix observation covariance matrix $\Sigma$, the minimizing weights solvewhere $\eta$ is a vector of Lagrange multipliers. Equivalently, the generalized least squares estimate and prediction areThe first term is the estimated trend alone; the second predicts the correlated regression residuals. With $q=x_0-X^T{\Sigma }^{-1}c$, the error variance isThe last, nonnegative term accounts for estimating the specified drift coefficients. Merely inserting a fitted drift into simple kriging and keeping its known-mean error variance would omit this term. For a Gaussian random field, the same expressions arise by integrating over drift coefficients in a diffuse prior distribution limit; this is an additional inferential interpretation, not an assumption required by the linear optimization.^[ref]