Asymptotics of kernel density estimators on weakly associated random fields

Let ZN, N≥1 denote the integer lattice points in the N-dimensional Euclidean space and {Xn} be an Rd-valued, strictly stationary, weakly associated random field indexed by ZN. Under mild conditions, as in [Roussas, G.G., 2000. Asymptotic normality of the kernel estimate of a probability density function under association. Statist. Probab. Lett. 50, 1-12], asymptotic normality of the kernel estimator of the multivariate density is established. We also investigate the mean squared error of the estimator towards the unknown density.