Optimal asymptotic quadratic errors of density estimators on random fields

Let Z(N) denote the integer lattice points in the N-dimensional Euclidean space. Kernel estimation of the multivariate density of a random field indexed by ZN is investigated. The loss between the estimator and the unknown density is measured by means of mean squared and mean integrated squared errors. Under mild mixing conditions, we show that the kernel density estimator has exactly the same asymptotic error as in the i.i.d. case. (C) 2002 Elsevier Science B.V. All rights reserved.