Central limit theorem for martingales.

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    In this note I discuss the most general known form of the central limit theorem for martingales and triangular arrays of martingale difference sequences. I present a slightly modified proof of the main result in B. M. Brown's work Martingale Central Limit Theorems in the journal The Annals of Mathematical Statistics (1971) volume 42 No.~1 59--66, which provides a slightly more general result and also briefly discuss Aryeh Dvoretzky's result in his work Asymptotic normality for sums of dependent random variables, in the II. volume of the Sixth Berkeley Symposium pages 513--535. He proved a result similar to that of Brown with the help of an essentially different method. I compare these two proofs, and also present the functional central limit theorem for triangular arrays of martingale difference sequences. The details of the proof of this result are not worked out. At the end of this note I discuss Lévy's characterization of Wiener processes which can be considered as a consequence of the central limit theorem for martingales.

    It is worth understanding not only the results and proofs discussed in this note, but also the ideas behind them. These results are proved with the help of the characteristic function technique. We estimate the characteristic functions by means of classical methods, but there is one point in the proof which deserves special attention. To get sharp results we have to work not with the variances of the terms in the sum we are investigating but with their conditional variances with respect to the past. This can be interpreted so that there appears some sort of `inner time' in the investigation, and this provides the natural time scaling. This shows some similarity with the behaviour of Ito integrals, and there is some hope that the method applied in this note may be useful also in the study of limit problems where limit is an Ito integral.

    22 pages