Random fields, stationarity, covariance, and variograms

Let $T\subseteq {\mathbb{R}}^d$, $d\ge 1$, have its relative Borel sigma-algebra, and let $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ be one probability space. A real random field indexed by $T$ is a family $X=(X(s){)}_{s\in T}$ of real random variables $X(s):\mathrm{\Omega }\to \mathbb{R}$ on that space. This is coordinate measurability: each map $(s,\omega )\mapsto X(s,\omega )$ with $s$ fixed is measurable in $\omega$. The field is jointly measurable when the map $(s,\omega )\mapsto X(s,\omega )$ from $(T\times \mathrm{\Omega },\mathcal{B}(T)\otimes \mathcal{F})$ to $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ is measurable. Joint measurability is an additional hypothesis when a spatial Lebesgue integral is used; it is not part of coordinate measurability.

Write $\mathcal{L}(X(s_1),\dots ,X(s_n))$ for the pushforward law of a finite coordinate tuple (Stochastic processes and their finite-dimensional distributions, Law or distribution of a random element). The field is strictly stationary on $T$ when, for every finite tuple $s_1,\dots ,s_n\in T$ and every $u\in {\mathbb{R}}^d$ for which $s_i+u\in T$ for all $i$, these joint laws are invariant under the common translation $s_i\mapsto s_i+u$. This extends the integer-shift convention for time series in Strict and weak stationarity for integer-indexed time series to spatial indices.

Write $T-T:=\{t-s:s,t\in T\}$ for the set of displacements realized by pairs of sites.

The field is wide stationary when every $X(s)$ is square-integrable, its mean is one constant $m$, and there is a function $C:T-T\to \mathbb{R}$ such that $\mathrm{Cov}(X(s),X(t))=C(t-s)(s,t\in T)\mathrm{.}$ For a wide-stationary field on all of ${\mathbb{R}}^d$, it is wide isotropic when $C(h)$ depends only on the Euclidean norm $\mathrm{\parallel }h\mathrm{\parallel }$.

For $h\in T-T$ and any $s$ with $s,s+h\in T$, put ${\mathrm{\Delta }}_{h}X(s):=X(s+h)-X(s)$. The field is intrinsically stationary of order two when for each $h\in T-T$ the first and second raw moments of ${\mathrm{\Delta }}_{h}X(s)$ are finite and independent of the admissible starting site $s$. Here these moments are $\mathbb{E}[{\mathrm{\Delta }}_{h}X(s)]$ and $\mathbb{E}[({\mathrm{\Delta }}_{h}X(s){)}^2]$. The common second moment defines the variogram $\gamma (h):=\frac{1}{2}\mathbb{E}[({\mathrm{\Delta }}_{h}X(s){)}^2]\mathrm{.}$ The mean increment may depend on $h$ and need not be zero; no finite second moment of the field levels themselves is required by this definition.

Spodarev's 2024 notes, §§1.1–1.2, state strict and wide invariance and define wide intrinsic stationarity through displacement-dependent increment first and second moments. The 2009 draft, §§1.3–1.4, uses a narrower zero-mean increment clause for intrinsic stationarity of order two; that extra restriction is not adopted. Both sources distinguish wide from strict stationarity. The covariance and variogram consequences proved on the A page are not supplied by this definition alone.