# Central limit theorems, Almost sure invariance principles and Brownian motion

In a paper I was reading on dynamics, I came across a proof of a central limit theorem in a certain situation using brownian motion and an almost sure invariance principle. I am not very experienced in probability theory; so I would like to know:

1. Is this a standard method to prove the central limit theorem?

2. What is possibly the advantage in such an approach?

3. What are some references to see simple instances of such a proof of CLT, law of iterated logarithm, etc?

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Could you further explain the kind of central limit theorem and the "certain situation using Brownian motion"? – Tim van Beek Sep 8 '11 at 13:01
@Tim van Beek: CLT as proved in this paper: maths.warwick.ac.uk/~mpollic/k.pdf – George Sep 8 '11 at 13:09

Or try the reference in the paper: 3] P. Billingsley, Convergence of probability measures, John Wiley, New York-London-Sydney 1968

It is actually a beautiful (but challenging) book

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The central limit theorem is generally proved by proving that the sequences of characteristic functions converge pointwise to a characteristic function of a normal random variable. This method is because the Lévy's Continuity Theorem that states:

A sequence $\{ X_j \}$ of $n$-variate random variables converges in distribution to random variable $X$ if and only if the sequence $\{ \varphi_{X_j} \}$ converges pointwise to a function $\varphi$ which is continuous at the origin. Then $\varphi$ is the characteristic function of $X$.

And the fact that a random variable is uniquely determined by its characteristic function.

The trick to proof the sequence converges is consider the expansion of Taylor of order $2$ of the characteristic functions and is pretty straightforward to see that converges to a characteristic function of a normal r.v.

The advantage is that the proof requires you only basic knowledge of calculus, if you consider true the uniqueness of the characteristic function and the theorem of Lévy.

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