Variogram

Remarks

Every variogram satisfies $\gamma(0) = 0$ and $\gamma(-h) = \gamma(h)$. It is conditionally negative definite: $\sum_{i,j} a_i a_j \gamma(s_i - s_j) \le 0$ whenever $\sum_i a_i = 0$. If $Z$ is weakly stationary with covariance function $C(h) = \operatorname{Cov}(Z(s + h), Z(s))$, then $\gamma(h) = C(0) - C(h)$.

Example: the standard Wiener process $(B(t))_{t \in \R}$, two-sided with $B(0) = 0$, has $B(t + h) - B(t)$ normal with mean $0$ and variance $\abs{h}$. So it is intrinsically stationary with $2\gamma(h) = \abs{h}$, although its variance $\abs{t}$ is not constant in $t$.

Non-example: let $X$ be a standard normal random variable and $Z(s) = s^2 X$ on $\R$. The increment $Z(s + h) - Z(s) = (2sh + h^2) X$ has mean $0$, but its variance $(2sh + h^2)^2$ depends on $s$, so clause 2 fails.