Kernel regression estimation for continuous spatial processes

We investigate here a kernel estimate of the spatial regression function r(x) = E(Y_uX_u = x), x ∈ ℝ^d, of a stationary multidimensional spatial process { Z_u = (X_u, Y_u), u ∈ ℝ^N}. The weak and strong consistency of the estimate is shown under sufficient conditions on the mixing coefficients and the bandwidth, when the process is observed over a rectangular domain of ℝ^N. Special attention is paid to achieve optimal and suroptimal strong rates of convergence. It is also shown that this suroptimal rate is preserved by using a suitable spatial sampling scheme.