Nonparametric prediction for random fields

We study prediction for vector valued random fields in a nonparametric setting. The prediction problem is formulated as the problem if estimating certain conditional expectations and a speed of uniform a.s. convergence is obtained, modifying results for conditional empirical processes derived from series with one-dimensional time. As an alternative to the usual mixing conditions we model the dependence by asymptotic decomposability. This includes linear (which generalizes ARMA) fields and random fields with a finite order Volterra expansion. As an example of a linear field we briefly discuss the finite-differences simulation of the heat equation blurred by additive random noise.