Asymptotic normality of kernel type regression estimators for random fields

The asymptotic normality of the Nadaraya-Watson regression estimator is studied for α-mixing random fields. The infill-increasing setting is considered, that is when the locations of observations become dense in an increasing sequence of domains. This setting fills the gap between continuous and discrete models. In the infill-increasing case the asymptotic normality of the Nadaraya-Watson estimator holds, but with an unusual asymptotic covariance structure. It turns out that this covariance structure is a combination of the covariance structures that we observe in the discrete and in the continuous case.