Central limit theorem for martingales.
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Series of problems in Probability Theory
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.
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