We study prediction for vector valued random fields in a nonparametric
setting. The prediction problem is formulated as the problem of estim
ating certain conditional expectations and a speed of uniform a.s. con
vergence is obtained, modifying results for conditional empirical proc
esses 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 exampl
e of a linear field we briefly discuss the finite-difference simulatio
n of the heat equation blurred by additive random noise.