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The family $(Y_x)$ is a Gaussian random field. For distinct $x,x^{\prime }$ its covariance isTwo distinct vertices of ${\mathbb{Z}}^d$ have a common neighbour exactly when their graph distance, equivalently their ${\ell }^1$ distance, is two. Since jointly Gaussian variables are independent exactly when they are uncorrelated,Thus the field has finite-range dependence, even though nearest-neighbour values are independent.

For $|\varphi |<1$, the bilateral sum ${\sum }_{k\in \mathbb{Z}}{\varphi }^{|k|}{z}^k$ equals $(1-{\varphi }^2)/((1-\varphi z)(1-\varphi z^{-1}))$ on the unit circle. Multiplying by ${\sigma }^2/(\pi (1-{\varphi }^2))$ gives the one-sided autoregressive spectrum and reads off its covariance coefficients.

For a scale-invariant primordial bispectrum, the displayed shape function is invariant under common rescaling of all three wavenumbers. It isolates triangle dependence from the overall power spectrum amplitude. It indicates intrinsic shape preference but alone does not determine observational signal-to-noise, which also depends on transfer functions, covariance and survey weights.

Quasi-likelihood specifies a mean and a variance function without necessarily specifying a full response distribution. For independent responses, the estimating equation is ${\sum }_i(\partial {\mu }_i/\partial \beta )(y_i-{\mu }_i)/(\varphi V({\mu }_i))=0$. A common scalar dispersion parameter rescales coefficient covariance while leaving the roots of the quasi-score equation unchanged.

A random slope multiplies a predictor by a group-specific latent coefficient. With $b_j\sim N(0,{\tau }^2)$, a contribution $b_j{t}_i$ induces covariance ${\tau }^2{t}_i{t}_k$ for two observations in group $j$. A random slope does not necessarily include a random intercept.

For a known zero mean, ${\hat{\gamma }}_n(h)=n^{-1}{\sum }_{t=1}^{n-h}{Y}_t{Y}_{t+h}$ for $0\le h<n$. Weakly stationary process assumptions give expectation $(1-h/n)\gamma (h)$. Dividing by $n-h$ instead gives an unbiased estimate at each individual lag, although the resulting collection need not form a positive semidefinite covariance sequence.

The sill is the large-distance level of a semivariogram when that limit exists. For covariance tending to zero, the sill is the total marginal variance; a nugget plus a structured component has total sill ${\tau }^2+{\sigma }^2$. A nonzero constant covariance component is invisible to the semivariogram.

For known zero mean, invertible observation covariance $\Sigma$, target covariance vector $c$ and target variance $v_0$, simple kriging predicts $c^T{\Sigma }^{-1}z$ with error variance $v_0-c^T{\Sigma }^{-1}c$. Completing the covariance quadratic form proves optimality. A multivariate normal distribution makes this the exact conditional mean and variance.

For a Gaussian free field with boundary heights $-\lambda$ and $+\lambda$ on the two sides of the starting point, its distinguished zero-height interface has chordal ${\mathrm{SLE}}_4$ law. Conditional on an initial curve segment, the field is a zero-boundary field in the slit domain plus $m_t(z)=\lambda -(2\lambda /\pi )\arg (g_t(z)-W_t)$. Its mean bracket equals minus the variation of its conditional Green covariance. Subtracting the initial harmonic function gives the corresponding coupling to a zero-boundary field.

Filtering multiplies the time-series spectral density by the squared modulus of the frequency response. Insert the spectral integral into the covariance double sum and interchange sums using absolute summability of the filter. This holds even when the input covariances are not absolutely summable, provided the spectral measure has a density.

This tensor integrates the local pressure tensor, where ${\sigma }_{ij}=⟨(v_i-u_i)(v_j-u_j)⟩$ is the velocity covariance. Half its trace is the random kinetic energy. It enters the tensor virial theorem with coefficient one.

If $Z$ is nondegenerate, $g$ is strictly increasing and the relevant products are integrable, an independent copy $Z^{\prime }$ gives $2\mathrm{Cov}(g(Z),Z)=\mathbb{E}[(g(Z)-g(Z^{\prime }))(Z-Z^{\prime })]$. The integrand is positive exactly when $Z\ne Z^{\prime }$, so the covariance is strictly positive. The same argument applies within each conditional law and yields a positive terminal stock hedge for an attainable increasing payoff.

A noise-driven barrier crossing between adjacent phase-locked wells. The Kramers escape rate is proportional to $e^{-\Delta V/T_{\mathrm{eff}}}$ when a mobility-one Langevin equation has white-noise covariance $2T_{\mathrm{eff}}\delta (t-t^{\prime })$. Both the barrier and local relaxation time determine whether an intrawell phase autocorrelation approximation is appropriate.

Place $n$ labelled balls independently into $m$ labelled bins, each with probability $1/m$. The vector of bin counts has a multinomial distribution, and each count separately has the binomial distribution $\mathrm{Bin}(n,1/m)$. Distinct bin counts have covariance $-n/m^2$, so they are dependent for $n>0$ and $m>1$. Counts of bins of a specified size are sums of indicator variables; their expectations follow from the linearity of expectation, without any independence between bins.

The variance of hypothetical means in the Bühlmann model is the between-risk variance of the conditional claim expected value. It creates the shared covariance $a$ between different years of the same risk and determines how much individual experience should influence the premium.

The velocity ellipsoid represents the positive velocity covariance tensor by its orthogonal principal axes and eigenvalue square roots. The uncentered second-moment tensor differs from covariance when a population streams. Vanishing mixed second moments align the second-moment ellipsoid with the chosen frame; for pure azimuthal streaming in a phase-mixed axisymmetric model, the off-diagonal covariance also vanishes.