Lagged lattice covariance and variogram estimators

Let $d\ge 1$, let $(Y_j{)}_{j\in {\mathbb{Z}}^d}$ be a real random field (Random fields, stationarity, covariance, and variograms) whose coordinates are square-integrable real random variables on one probability space (Probability measures and probability spaces, Random elements and real random variables, Moments, variance, and covariance on a probability space, The integers as equivalence classes of pairs of naturals), and let $V_n\subset {\mathbb{Z}}^d$ be nonempty finite observation windows. Write $M_n:=\mathrm{\mid }{V}_n\mathrm{\mid }$ and ${\bar{Y}}_n:=\frac{1}{M_n}{\sum }_{j\in V_n}{Y}_j\mathrm{.}$ For a fixed lag $h\in {\mathbb{Z}}^d$, let $P_n(h):=\{j\in V_n:j+h\in V_n\},N_n(h):=\mathrm{\mid }{P}_n(h)\mathrm{\mid }\mathrm{.}$ Only indices $n$ with $N_n(h)>0$ are admissible for this lag.

When the field is wide stationary in the sense of Random fields, stationarity, covariance, and variograms, write its constant mean as $\mu$ and its displacement covariance as $C(h):=\mathrm{Cov}(Y_{j+h},Y_j)$, which is independent of $j$ by wide stationarity and agrees with the supplier convention by covariance symmetry. Define the pair-count-normalized raw-product covariance estimator by ${\widehat{C}}_n(h):=\frac{1}{N_n(h)}{\sum }_{j\in P_n(h)}{Y}_j{Y}_{j+h}-{\bar{Y}}_n^2\mathrm{.}$ This convention subtracts the square of the one global sample mean from the pair-count-normalized raw product average; it is not the average of pairwise sample-centered products.

For the second-moment component of order-two intrinsic stationarity in Random fields, stationarity, covariance, and variograms, suppose the increments have finite second moments and $\gamma (h):=\frac{1}{2}\mathbb{E}[(Y_{j+h}-Y_j{)}^2]$ is independent of $j$; define the empirical semivariogram by ${\widehat{\gamma }}_n(h):=\frac{1}{2N_n(h)}{\sum }_{j\in P_n(h)}(Y_{j+h}-Y_j{)}^2\mathrm{.}$