Histograms for stationary linear random fields

Denote the integer lattice points in the $$N$$ N -dimensional Euclidean space by $$\mathbb {Z}^N$$ Z N and assume that $$X_\mathbf{n}$$ X n , $$\mathbf{n} \in \mathbb {Z}^N$$ n ∈ Z N is a linear random field. Sharp rates of convergence of histogram estimates of the marginal density of $$X_\mathbf{n}$$ X n are obtained. Histograms can achieve optimal rates of convergence $$({\hat{\mathbf{n}}}^{-1} \log {\hat{\mathbf{n}}})^{1/3}$$ ( n ^ - 1 log n ^ ) 1 / 3 where $${\hat{\mathbf{n}}}=n_1 \times \cdots \times n_N$$ n ^ = n 1 × ⋯ × n N . 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