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We prove a central limit theorem for stationary multiple (random) fields ofmartingale differences $f\circ T_{\underline{i}}$, $\underline{i}\in \Bbb Z^d$,where $T_{\underline{i}}$ is a $\Bbb Z^d$ action. In most cases the multiple(random) fields of martingale differences is given by a completely commutingfiltration. A central limit theorem proving convergence to a normal law hasbeen known for Bernoulli random fields and in [V15] this result was extended torandom fields where one of generating transformations is ergodic. In thepresent paper it is proved that a convergence takes place always and the limitlaw is a mixture of normal laws. If the $\Bbb Z^d$ action is ergodic and $d\geq2$, the limit law need not be normal. For proving the result mentioned above, ageneralisation of McLeish's CLT for arrays $(X_{n,i})$ of martingaledifferences is used. More precisely, sufficient conditions for a CLT are foundin the case when the sums $\sum_i X_{n,i}^2$ converge only in distribution. TheCLT is followed by a weak invariance principle. It is shown that central limittheorems and invariance principles using martingale approximation remain validin the non-ergodic case.

On limit theorems for fields of martingale differences