Unbiasedness and L2 consistency of the spatial window average

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.