# Central Limit Theorem

This week, in the MAT8595 course, before proving Fisher-Tippett theorem, we will get back on the proof of the Central Limit Theorem, and the class of stable distribution (in Lévy’s sense). In order to illustrate the problem of heavy tails, on the behavior of the mean, consider a sequence of i.i.d. Gaussian random variables $X_i$‘s. On top, we visualize the sequence, and below, we visualize the associate random walk

$S_n=\sum_{i=1}^n X_i$

(the central limit theorem will give a limiting distribution for $n^{-1}S_n$ in the case where the variance of the $X_i$‘s is finite)

If we consider a sequence of i.i.d. random variables $X_i$‘s whith heavier tails (possibly with infinite variance), we can still define $S_n$, but as we can see below, $S_n$ can be quite heratic.

As we will see this Thursday, the key to derive stable distribution for the central limit theorem, or possible limiting distributions for the maximum is Cauchy’s function equation. I strongly recommand to look at the proof.