On the Functional Central Limit Therorem for Martingales

Necessary and sufficient conditions for the functional central limit theorem for a double array of random variables are sought. It is argued that this is a martingale problem only if the variables truncated at some fixed point c are asymptotically a martingale difference array. Under this hypothesis, necessary and sufficient conditions for convergence in distribution to a Brownian motion are obtained when the normalization is given (i) by the suns of squares of the variables, (ii) by the conditional variances and (iii) by the variances. The results are proved by comparing the various riorralizations with s "natural" normalization. KB 1970 Subject classifications: Primary 60B10; Secondary 60G45.