An equivalent of martingale approximation for random fields

      The method of martingale approximation consists in approximating the partial sums of a strictly stationary sequence by those of a martingale differences sequence with stationary increments. Gordin and Peligrad (2011) gave a necessary and sufficient condition for martingale approximation. This condition can be used to establish an invariance principle. After having reviewed the main results in dimension one, we will present the extension to processes indexed by $\mathbb Z^d$, $d\geqslant 1$.