The Classical Central Limit Theorem

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Let $\{X_n\}_{n=1}^\infty$ be a sequence of i.i.d. random variables with mean $m$ and finite variance $\sigma^2>0$ and define $S_n=\sum_{j=1}^nX_n$.  Then $$ \frac{S_n-mn}{\sigma\sqrt{n}}\overset{L}{\longrightarrow} \Phi. $$

$\bullet$ Proof.

To prove the Classical CLT, we use the characteristic function of $X_j$, $f(t)$.  Without loss of generality, let $m=0$. we have $$\begin{array}{ccl}
\mathscr{E}\left(\exp\left\{it\frac{S_n}{\sigma\sqrt{n}}\right\}\right)
& = & f\left(\frac{t}{\sigma\sqrt{n}}\right)^n \\
& = & \left\{1+\frac{i^2\sigma^2}{2}\left(\frac{t}{\sigma\sqrt{n}}\right)^2+o\left(\frac{|t|}{\sigma\sqrt{n}}\right)^2\right\}^n \\
& = & \left\{1-\frac{t^2}{2n}+o\left(\frac{t^2}{n}\right)\right\}^n \overset{n}{\longrightarrow} \exp\left\{-\frac{t^2}{2}\right\},  \end{array}$$ which is the ch.f. of $\Phi$.

$\Box$