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  "body": "\n## Definition\n\nLet $D = \\R^d$ and let $(Z(s))_{s \\in D}$ be a real random field with $E[Z(s)^2] \u003c \\infty$ for all $s$.\nThe field is *intrinsically stationary* if for every $h \\in \\R^d$:\n\n1. $E[Z(s + h) - Z(s)] = 0$ for all $s$, and\n2. $\\operatorname{Var}(Z(s + h) - Z(s))$ does not depend on $s$.\n\nFor such a field the *variogram* is the function $2\\gamma \\colon \\R^d \\to [0, \\infty)$ given by\n$$2\\gamma(h) = \\operatorname{Var}(Z(s + h) - Z(s)) = E[(Z(s + h) - Z(s))^2],$$\nand $\\gamma$ is the *semivariogram*.\n\n## Remarks\n\nEvery variogram satisfies $\\gamma(0) = 0$ and $\\gamma(-h) = \\gamma(h)$.\nIt is conditionally negative definite: $\\sum_{i,j} a_i a_j \\gamma(s_i - s_j) \\le 0$ whenever $\\sum_i a_i = 0$.\nIf $Z$ is weakly stationary with [[covariance|0KBFAWYRTE]] function $C(h) = \\operatorname{Cov}(Z(s + h), Z(s))$, then $\\gamma(h) = C(0) - C(h)$.\n\nExample: the standard [[Wiener process|22735FGJV6]] $(B(t))_{t \\in \\R}$, two-sided with $B(0) = 0$, has $B(t + h) - B(t)$ normal with mean $0$ and variance $\\abs{h}$.\nSo it is intrinsically stationary with $2\\gamma(h) = \\abs{h}$, although its variance $\\abs{t}$ is not constant in $t$.\n\nNon-example: let $X$ be a standard normal random variable and $Z(s) = s^2 X$ on $\\R$.\nThe 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.\n",
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