Asymptotic properties of nonparametric regression for long memory random fields

The kernel estimator of spatial regression function is investigated for stationary long memory (long range dependent) random fields observed over a finite set of spatial points. A general result on the strong consistency of the kernel density estimator is first obtained for the long memory random fields, and then, under some mild regularity assumptions, the asymptotic behaviors of the regression estimator are established. For the linear long memory random fields, a weak convergence theorem is also obtained for kernel density estimator. Finally, some related issues on the inference of long memory random fields are discussed through a simulation example.