## The Central Limit Theorem

[latexpage]The proof and intuition presented here come from this excellent writeup by Yuval Filmus, which in turn draws upon ideas in this book by Fumio Hiai and Denes Petz. Suppose that we have a sequence of real-valued random variables

X_1, X_2, \ldots .

Define the random variable

A_N = \frac{X_1 + \cdots + X_N}{\sqrt{N}}

to be a scaled sum of the first $N$ variables in the sequence. Now, we would like to make interesting statements about the sequence

A_1, A_2, \ldots .