  We prove a central limit theorem for stationary multiple (random) fields of
martingale 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 commuting
filtration. A central limit theorem proving convergence to a normal law has
been known for Bernoulli random fields and in [V15] this result was extended to
random fields where one of generating transformations is ergodic. In the
present paper it is proved that a convergence takes place always and the limit
law is a mixture of normal laws. If the $\Bbb Z^d$ action is ergodic and $d\geq
2$, the limit law need not be normal. For proving the result mentioned above, a
generalisation of McLeish's CLT for arrays $(X_{n,i})$ of martingale
differences is used. More precisely, sufficient conditions for a CLT are found
in the case when the sums $\sum_i X_{n,i}^2$ converge only in distribution. The
CLT is followed by a weak invariance principle. It is shown that central limit
theorems and invariance principles using martingale approximation remain valid
in the non-ergodic case.
