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Motivated by random evolutions which do not start from equilibrium, in arecent work, Peligrad and Voln\'{y} (2018) showed that the quenched CLT(central limit theorem) holds for ortho-martingale random fields. In thispaper, we study the quenched CLT for a class of random fields larger than theortho-martingales. To get the results, we impose sufficient conditions in termsof projective criteria under which the partial sums of a stationary randomfield admit an ortho-martingale approximation. More precisely, the sufficientconditions are of the Hannan's projective type. As applications, we establishquenched CLT's for linear and nonlinear random fields with independentinnovations.

On the quenched CLT for stationary random fields under projective criteria