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:


*

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

*What is possibly the advantage in such an approach?

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