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For a random field or Stochastic process $$Z(x)$$ on a domain $$D$$, a covariance function $$C(x,y)$$ gives the covariance of the values of the random field at the two locations $$x$$ and $$y$$:

$$C(x,y):=Cov(Z(x),Z(y)).$$

The same $$C(x,y)$$ is called autocovariance in two instances: in time series (to denote exactly the same concept, but where $$x$$ is time), and in multivariate random fields (to refer to the covariance of a variable with itself, as opposed to the cross covariance between two different variables at different locations, $$Cov[Z(x_1),Y(x_2)]$$).

Admissibilty
For locations $$x_1, x_2, \ldots, x_N \in D$$ the variance of every linear combinations

$$X=\sum_{i=1}^N w_i Z(x_i)$$

can be computed by

$$var(X)=\sum_{i=1}^N \sum_{i=1}^N w_i C(x_i,x_j) w_j$$

A function is a valid covariance function if and only if this variance is non-negative for all possible choices of N and weights $$w_1,\ldots,w_N$$. A function with this property is called positive definite.

Simplifications with Stationarity
In case of a second order stationary random field, where

$$C(x_i,x_j)=C(x_i+h,x_j+h)$$

for any lag $$h$$, the covariance function can represented by a one parameter function

$$C_s(h)=C(0,h)=C(x,x+h)$$

which is called covariogram or also covariance function. Implicitly the $$C(x_i,x_j)$$ can be computed from $$C_s(h)$$ by:

$$C(x,y)=C_s(y-x)$$

The positive definitness of the single argument version of the covariance function can be checked by Bochner's theorem.