On the Functional Central Limit Theorem for Martingales
NORTH CAROLINA UNIV AT CHAPEL HILL DEPT OF STATISTICS
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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 sums of squares of the variables, ii by the conditional variances and iii by the variances. The results are proved by comparing the various normalizations with a natural normalization.
- Statistics and Probability