Consistency of a nonparametric conditional mode estimator for random fields

Given a stationary multidimensional spatial process $left{ Z_{mathbf{i}}=left( X_{mathbf{i}}, Y_{mathbf{i}}right) in mathbb R ^dright. left. times mathbb R ,mathbf{i}in mathbb Z ^{N}right} $ , we investigate a kernel estimate of the spatial conditional mode function of the response variable $Y_{mathbf{i}}$ given the explicative variable $X_{mathbf{i}}$ . Consistency in $L^p$ norm and strong convergence of the kernel estimate are obtained when the sample considered is a $alpha $ -mixing sequence. An application to real data is given in order to illustrate the behavior of our methodology.