Stationarity of Order 2

A random function is said to be stationary of order 2, if:

(i)

the mathematical expectation $E[Z({/boma x})]$ exists and does not depend on the support point ${/boma x}$; thus,

(ii)

for each pair of random variables $/{Z({/boma x}),Z({/boma x}+{/boma h})/}$ the covariance exists and depends on the separation distance ${/boma h}$,

The stationarity of the covariance implies the stationarity of the variance and the variogram. The following relations are immediately evident:

The last equation indicates that, under the hypothesis of second-order stationarity, the covariance and the variogram are two equivalent tools for characterizing the auto-correlations between two variables $Z({/boma x}+{/boma h})$ and $Z({/boma x})$ separated by a distance ${/boma h}$. We can also define a third tool, the correlogram:

The hypothesis of second-order stationarity assumes the existence of a covariance and, thus, of a finite a priori variance, $Var[Z({/boma x})] = C({/boma 0})$. Now, the existence of the variogram function represents a weaker hypothesis than the existence of the covariance; moreover, there are many physical phenomena and random variables which have an infinite capacity for dispersion, i.e., which have neither an a priori variance nor a covariance, but for which a variogram can be defined. As a consequence, the second-order stationarity hypothesis can be slightly reduced (weakened) if assuming only the existence and stationarity of the variogram.