Spatial summary estimators

Given: $d\ge 1$, a probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$, a jointly measurable real field $X:{\mathbb{R}}^d\times \mathrm{\Omega }\to \mathbb{R}$, $\mathbb{E}[X(s)]=\mu$, $\mathbb{E}[X(s{)}^2]<\mathrm{\infty }$ for every $s$, stationary covariance $C(t-s)$, ${\int }_{{\mathbb{R}}^d}\mathrm{\mid }C(h)\mathrm{\mid }dh<\mathrm{\infty }$, and a Van Hove sequence $(W_n)$.

Van Hove sequences of spatial windows: under ${\mathrm{A}\mathrm{C}}_{\omega }$, Assuming countable choice, every Borel subset of ${\mathbb{R}}^n$ is Lebesgue measurable makes every Borel $W_n$ Lebesgue measurable and Assuming countable choice, $\mathcal{L}({\mathbb{R}}^n)$ is a sigma-algebra containing every elementary set and ${\lambda }_n$ is a complete measure extending elementary volume supplies Lebesgue measure; every $W_n$ has finite positive volume, $\mathrm{\mid }{W}_n\mathrm{\mid }\to \mathrm{\infty }$, and for each compact $K$, $\mathrm{\mid }(W_n+K)\setminus W_n\mathrm{\mid }\mathrm{/}\mathrm{\mid }{W}_n\mathrm{\mid }\to 0$.

Spatial window average: when a sample path is Lebesgue integrable, ${\bar{X}}_{W_n}=\mathrm{\mid }{W}_n{\mathrm{\mid }}^{-1}{\int }_{W_n}X(s)ds$.

Random fields, stationarity, covariance, and variograms, Random elements and real random variables, and The Borel sigma-algebra of a topological space: a field is a family of real random variables on one probability space; joint measurability is an additional hypothesis for spatial integration, and wide stationarity supplies a constant mean and displacement-indexed covariance.

Moments, variance, and covariance on a probability space: variance is the expected centered square and covariance is the expected centered product; symmetry makes the reverse-lag convention in the Given equivalent to the supplier covariance convention.

Expectation of a nonnegative or integrable random variable: expectation is the Lebesgue integral against $\mathbb{P}$.

Cauchy-Schwarz for random variables: for $U,V\in L^2(\mathbb{P})$, $\mathbb{E}\mathrm{\mid }UV\mathrm{\mid }\le \mathrm{\parallel }U{\mathrm{\parallel }}_2\mathrm{\parallel }V{\mathrm{\parallel }}_2$.

The product measure of two sigma-finite measure spaces and Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: nonnegative product- measurable functions can be integrated in either order, and the section integral is measurable.

Integrable real and complex functions, and their integrals and Fubini's theorem for L^1 functions on a sigma-finite product: absolutely integrable product-measurable functions have equal iterated and product integrals.

Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation: Lebesgue measure is invariant under spatial translations.

$L^p$ convergence for random variables: $L^2$ convergence means that the $L^2$ norm of the difference tends to zero.

The product sigma-algebra and its finite iterates, A measurable function between measurable spaces, Every section of a product-measurable function is measurable, and Composition with a Borel measurable outer map preserves measurability: on $({\mathbb{R}}^d\times {\mathbb{R}}^d\times \mathrm{\Omega },(\mathcal{B}({\mathbb{R}}^d)\otimes \mathcal{B}({\mathbb{R}}^d))\otimes \mathcal{F})$, the maps $((s,t),\omega )\mapsto (s,\omega )$ and $((s,t),\omega )\mapsto (t,\omega )$ are measurable because the preimages of generating rectangles $A\times E$ are respectively $(A\times {\mathbb{R}}^d)\times E$ and $({\mathbb{R}}^d\times A)\times E$. Composing with the jointly measurable field shows that $(s,t,\omega )\mapsto X(s,\omega )$ and $(s,t,\omega )\mapsto X(t,\omega )$ are measurable, so their centered product is measurable. Also, $(s,h)\mapsto s+h$ is continuous because $\mathrm{\parallel }s+h-(s_0+h_0){\mathrm{\parallel }}_2\le \mathrm{\parallel }s-s_0{\mathrm{\parallel }}_2+\mathrm{\parallel }h-h_0{\mathrm{\parallel }}_2$. Every open subset of ${\mathbb{R}}^d\times {\mathbb{R}}^d$ is a countable union of rational open rectangles, hence belongs to $\mathcal{B}({\mathbb{R}}^d)\otimes \mathcal{B}({\mathbb{R}}^d)$. Thus inverse images of open sets under addition are product-measurable, and the class of sets with product-measurable preimage is a sigma-algebra containing all open sets, hence contains every Borel set. Therefore addition is measurable for the product sigma-algebra and $(s,h)\mapsto {\mathbf{1}}_{W_n}(s){\mathbf{1}}_{W_n}(s+h)$ is measurable for Borel $W_n$. Under AC${}_{\omega }$ in [A1], these Borel-measurable functions are also measurable for the product Lebesgue sigma-algebras used in Tonelli. Since $C$ is Lebesgue measurable by $C\in L^1$, its lift $(s,h)\mapsto C(h)$ is measurable for the product Lebesgue sigma-algebra: the preimage of a Lebesgue-measurable set $E$ is ${\mathbb{R}}^d\times E$. Thus $C(h){\mathbf{1}}_{W_n}(s){\mathbf{1}}_{W_n}(s+h)$ is product-measurable as well.

Dominated convergence: pointwise convergence dominated by an integrable function permits passage of the limit through the integral.

Lebesgue measure is sigma-finite, and every metrically bounded subset of ${\mathbb{R}}^n$ has finite outer measure: under the already assumed ${\mathrm{A}\mathrm{C}}_{\omega }$, Lebesgue measure on ${\mathbb{R}}^d$ is sigma-finite, as witnessed by its increasing finite-measure cubes; its restriction to each bounded measurable $W_n$ is finite.

Given: The lattice field, windows, fixed lag, and $L^2$-consistent sample means in the Statement.

Lagged lattice covariance and variogram estimators: for $P_n(h)=\{j\in V_n:j+h\in V_n\}$ and $N_n(h)=\mathrm{\mid }{P}_n(h)\mathrm{\mid }$, ${\widehat{C}}_n(h)=\frac{1}{N_n(h)}{\sum }_{j\in P_n(h)}{Y}_j{Y}_{j+h}-{\bar{Y}}_n^2,$ and ${\widehat{\gamma }}_n(h)=\frac{1}{2N_n(h)}{\sum }_{j\in P_n(h)}(Y_{j+h}-Y_j{)}^2\mathrm{.}$

Moments, variance, and covariance on a probability space: for square-integrable real random variables, covariance is the expectation of the centered product.

Linearity, monotonicity, and the modulus bound for expectation: expectation is linear on finite sums of integrable variables, and $\mathrm{\mid }\mathbb{E}[Z]\mathrm{\mid }\le \mathbb{E}[\mathrm{\mid }Z\mathrm{\mid }]$ for integrable $Z$.

Cauchy-Schwarz for random variables: for square-integrable $U,V$, $\mathbb{E}[\mathrm{\mid }UV\mathrm{\mid }]\le \mathrm{\parallel }U{\mathrm{\parallel }}_2\mathrm{\parallel }V{\mathrm{\parallel }}_2<\mathrm{\infty }$, so $UV$ is integrable.

$L^p$ convergence for random variables: ${\bar{Y}}_n\to \mu$ in $L^2$ means $\mathrm{\parallel }{\bar{Y}}_n-\mu {\mathrm{\parallel }}_2\to 0$.

Given: An integer $d\ge 1$, coordinatewise addition on ${\mathbb{Z}}^d$, and the discrete topology.

The integers form a commutative ring and Group and abelian group: addition on $\mathbb{Z}$ is an abelian group operation; in coordinates, the same associativity, commutativity, zero, and inverse laws hold on ${\mathbb{Z}}^d$.

The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies: in the discrete topology every subset is open.

The product set ${\prod }_{i\in I}{X}_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space: products have the product topology, whose basic open sets restrict coordinates by open subsets.

Continuity of a map of topological spaces at a point and globally: a map is continuous when its open-set preimages are open.

Topological group: multiplication and inversion are continuous: a group with a topology is a topological group when multiplication and inversion are continuous.

Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not: distinct points must have disjoint open neighbourhoods to be Hausdorff.

Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right: every finite topological space, in particular a singleton, is compact.

Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space: a space is locally compact when each point has a compact neighbourhood.

Given: $d\ge 1$, the discrete additive group ${\mathbb{Z}}^d$, the multiplicative unit circle $\mathbb{T}$, and the map $\mathrm{\Psi }$ in the Statement.

The Pontryagin dual with the compact-open topology: a character is a continuous group homomorphism to $\mathbb{T}$, and the dual consists of these characters with pointwise multiplication and the compact-open topology.

On a discrete domain the compact-open topology is the topology of pointwise convergence: on a discrete domain, the compact-open topology on the character set is the subspace topology inherited from ${\mathbb{T}}^{{\mathbb{Z}}^d}$; in particular, evaluation at any fixed lattice point is continuous.

The integers as equivalence classes of pairs of naturals, The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies, Monoid homomorphism and group homomorphism, Integer powers in the complex field, and Exponent laws in a group: $g^{m+n}=g^m{g}^n$ and $(g^m{)}^n=g^{mn}$ for all $m,n\in \mathbb{Z}$, and $(gh{)}^n=g^n{h}^n$ when $g$ and $h$ commute: ${\mathbb{Z}}^d$ has the discrete topology, each $k\in {\mathbb{Z}}^d$ has integer coordinates, a homomorphism preserves addition, and integer powers satisfy $z^{a+b}=z^a{z}^b$ for $z\in \mathbb{T}$.

The multiplicative unit circle is a compact metrizable topological abelian group and Topological group: multiplication and inversion are continuous and A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice: $\mathbb{T}$ is a topological abelian group, integer-power maps on it are continuous by repeated multiplication and inversion, and a map into a product is continuous when each coordinate map is continuous.

Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological: a homeomorphism is a bijection that is continuous with continuous inverse.

The integer lattice is a discrete locally compact abelian group: ${\mathbb{Z}}^d$ with coordinatewise addition and the discrete topology is a Hausdorff locally compact abelian topological group; maps out of it are continuous.

Given: The homeomorphism $\mathrm{\Psi }$ and the finite positive Radon measure $\mu$ from the Statement.

The Pontryagin dual of the integer lattice is the torus: $\mathrm{\Psi }$ is a homeomorphism, so it and its inverse preserve Borel measurability, compactness and openness.

Radon measure on an LCH space: a Radon measure is finite on compact sets, outer regular on Borel sets and inner regular on open sets.

The Lebesgue integral is linear on $L^1(\mu )$, Every nonnegative measurable function is the increasing limit of simple measurable functions, and Monotone convergence for the integral: integrals of simple functions are linear, every nonnegative measurable function is the increasing limit of nonnegative measurable simple functions, and increasing limits pass through the integral.

The multiplicative unit circle is a compact metrizable topological abelian group, A product of finitely many compact spaces is compact in the product topology, and Arbitrary products preserve $T_0$, $T_1$, and Hausdorffness: ${\mathbb{T}}^d$ is compact Hausdorff, hence locally compact Hausdorff.

Integrable real and complex functions, and their integrals: bounded Borel functions are integrable against the finite measures in the Statement.

Given: The lattice field, constant mean, covariance convention, and periodogram definition in the Statement; assume AC and DC.

The Axiom of Choice: the Axiom of Choice is assumed.

The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain: Dependent Choice is assumed.

Positive definite functions on an abelian group: a function $c:G\to \mathbb{C}$ is positive definite when every finite quadratic form ${\sum }_{r,s}{a}_r\overline{a_s}c(x_r-x_s)$ is nonnegative.

Bochner's theorem for LCA groups: under AC and DC, every continuous positive-definite function on an LCA group has a unique finite positive Radon representing measure on its Pontryagin dual.

The Pontryagin dual of the integer lattice is the torus: $\widehat{{\mathbb{Z}}^d}$ is identified with the product torus by ${\chi }_{\theta }(j)=e^{ij\cdot \theta }$.

Transport of a lattice spectral measure to torus coordinates: a finite positive Radon measure on $\widehat{{\mathbb{Z}}^d}$ transports to a finite positive Radon measure on ${\mathbb{T}}^d$ while preserving integrals after composition.

Regular-grid spatial periodogram and smoothed estimator: for ${\mathrm{\Lambda }}_n=\{0,\dots ,n-1{\}}^d$ and $N_n=n^d$, $I_n(\lambda )=\frac{1}{(2\pi {)}^d{N}_n}{\mid {\sum }_{j\in {\mathrm{\Lambda }}_n}(Y_j-\mu )e^{-ij\cdot \lambda }\mid }^2\mathrm{.}$

Moments, variance, and covariance on a probability space: covariance is the expected centered product of square-integrable real random variables.

Linearity, monotonicity, and the modulus bound for expectation and Cauchy-Schwarz for random variables: expectation is linear on finite integrable sums, and products of square-integrable variables are integrable.

The product measure of two sigma-finite measure spaces and Fubini's theorem for L^1 functions on a sigma-finite product: the product angular Lebesgue measure exists and integrals can be evaluated by sections.

The integer lattice is a discrete locally compact abelian group: ${\mathbb{Z}}^d$ with the discrete topology is Hausdorff, abelian and locally compact, and every function on it is continuous.

Integer exponentials are orthogonal for angular Lebesgue measure: under Countable Choice, for every integer $k$, ${\int }_{-\pi }^{\pi }{e}^{ikx}dx$ is $2\pi$ when $k=0$ and zero otherwise.

AC implies DC implies countable choice: AC implies Countable Choice (through DC).

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): Countable Choice selects one element from each set in a countable family of nonempty sets.

Given: AC and DC, the wide-stationary continuous-space field with its specified integrable nonnegative density, and fixed regular-grid spacing $\delta >0$.

Random fields, stationarity, covariance, and variograms: wide stationarity gives a constant mean and displacement covariance; restricting to sites $\delta j$ preserves these properties.

A linear map $T$ of ${\mathbb{R}}^n$ sends Lebesgue measurable sets to Lebesgue measurable sets, with ${\lambda }_n(T[E])=\mathrm{\mid }\det T\mathrm{\mid }{\lambda }_n(E)$ when $T$ is invertible and $T[E]$ Lebesgue null when it is not and Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation: a scalar dilation by $\delta$ multiplies $d$-dimensional Lebesgue measure by ${\delta }^d$, and translations preserve it. The dilation theorem assumes Countable Choice.

Monotone convergence for the integral, Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Fubini's theorem for L^1 functions on a sigma-finite product, and Integrable real and complex functions, and their integrals: nonnegative countable sums pass through integrals, and absolutely integrable complex integrands can be partitioned and integrated termwise.

AC implies DC implies countable choice: AC implies DC and Countable Choice. The Axiom of Choice The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain

The multiplicative unit circle is a compact metrizable topological abelian group, A product of finitely many compact spaces is compact in the product topology, and Locally finite Borel measures on second-countable LCH spaces are regular: the finite-dimensional torus is compact Hausdorff and second-countable, and under Countable Choice its finite Borel measures are Radon. A countable base is obtained by intersecting rational Euclidean balls with its realization in ${\mathbb{R}}^{2d}$.

Fejer convolution formula for the lattice periodogram: under AC and DC a stationary lattice covariance has a unique finite positive Radon torus spectral measure, and $E{I}_n$ is its Fejer convolution.

The integers as equivalence classes of pairs of naturals and The integer-defining relation is an equivalence relation: $\mathbb{Z}$ is the quotient of $\mathbb{N}\times \mathbb{N}$ by the stated equivalence relation. Let $\sim$ be an equivalence relation on $A$ with quotient map $\pi :A\to A\mathrm{/}\sim$, and let $f:A\to B$. There is a function $g:A\mathrm{/}\sim \to B$ with $g\circ \pi =f$ if and only if $a\sim a^{\mathrm{\prime }}$ implies $f(a)=f(a^{\mathrm{\prime }})$; and such a $g$ is then unique: the map sending a pair to its class is surjective. $\mathbb{N}\times \mathbb{N}\approx \mathbb{N}$ supplies a bijection $\beta :\mathbb{N}\to \mathbb{N}\times \mathbb{N}$. A nonempty set is at most countable iff it is a surjective image of $\mathbb{N}$: a nonempty set is at most countable if and only if it admits a surjection from $\mathbb{N}$. Every finite power of an at most countable set is at most countable: every finite power of an at most countable set is at most countable.