On the functional central limit theorem for martingales |
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Authors: | Rootzén Holger |
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Institution: | (1) Department of Mathematical Statistics, University of Lund and Lund Institute of Technology, Box 725, S-22007 Lund, Sweden |
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Abstract: | 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.Research sponsored in part by the Office of Naval Research, Contract N00014-75-C-0809 |
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