[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 \begin{equation} X_1, X_2, \ldots . \end{equation} Define the random variable \begin{equation} A_N = \frac{X_1 + \cdots + X_N}{\sqrt{N}} \end{equation} to be a scaled sum of the first $N$ variables in the sequence. Now, we would like to make interesting statements about the sequence \begin{equation} A_1, A_2, \ldots . \end{equation}