Histograms for stationary linear random fields

Denote the integer lattice points in the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$N$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>N</mi> </math> </EquationSource> </InlineEquation>-dimensional Euclidean space by <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\mathbb {Z}^N$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>N</mi> </msup> </math> </EquationSource> </InlineEquation> and assume that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$X_\mathbf{n}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>X</mi> <mi mathvariant="bold">n</mi> </msub> </math> </EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$\mathbf{n} \in \mathbb {Z}^N$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="bold">n</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>N</mi> </msup> </mrow> </math> </EquationSource> </InlineEquation> is a linear random field. Sharp rates of convergence of histogram estimates of the marginal density of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$X_\mathbf{n}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>X</mi> <mi mathvariant="bold">n</mi> </msub> </math> </EquationSource> </InlineEquation> are obtained. Histograms can achieve optimal rates of convergence <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$({\hat{\mathbf{n}}}^{-1} \log {\hat{\mathbf{n}}})^{1/3}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mover accent="true"> <mi mathvariant="bold">n</mi> <mo stretchy="false">^</mo> </mover> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>log</mo> <mover accent="true"> <mi mathvariant="bold">n</mi> <mo stretchy="false">^</mo> </mover> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>3</mn> </mrow> </msup> </math> </EquationSource> </InlineEquation> where <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$${\hat{\mathbf{n}}}=n_1 \times \cdots \times n_N$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mover accent="true"> <mi mathvariant="bold">n</mi> <mo stretchy="false">^</mo> </mover> <mo>=</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <msub> <mi>n</mi> <mi>N</mi> </msub> </mrow> </math> </EquationSource> </InlineEquation>. The assumptions involved can easily be checked. Histograms appear to be very simple and good estimators from the point of view of uniform convergence. Copyright Springer Science+Business Media Dordrecht 2014