  In this paper we study the central limit theorem and its functional form for
random fields which are not started from their equilibrium, but rather under
the measure conditioned by the past sigma field. The initial class considered
is that of orthomartingales and then the result is extended to a more general
class of random fields by approximating them, in some sense, with an
orthomartingale. We construct an example which shows that there are
orthomartingales which satisfy the CLT but not its quenched form. This example
also clarifies the optimality of the moment conditions used for the validity of
our results. Finally, by using the so called orthomartingale-coboundary
decomposition, we apply our results to linear and nonlinear random fields.
