Kernel density estimation for stationary random fields

This paper establishes the asymptotic normality of the Parzen-Rosenblatt density estimator for stationary random fields under natural and easily verifiable conditions. We deal with random fields of the form $X_k = g(\epsilon_{k-s}, s \in \Z^d)$, $k\in\Z^d$, where $(epsilon_i)_{i\in\Z^d}$ are i.i.d random variables and $g$ is a measurable function. Such kind of spatial processes provides a general framework for stationary ergodic random fields. In particular, in the one-dimensional case, this class of processes includes linear as well as many widely used nonlinear time series models as special cases.