Central Limit Theorem

Definition

Let $X_1, X_2,\cdots, X_n$ be i.i.d. random variables with expected value $\mathbb{E}[X_i]=\mu < \infty$ and variance $0<\mathbb{Var}(X_i)=\sigma^2<\infty$. Then, the random variable

$$\begin{align*} Z_n=\frac{\sqrt{n}(\bar{X}_n - \mu)}{\sigma}&=\frac{\sqrt{n}}{\sigma}\Big(\frac{1}{n}\sum_{i=1}^nX_i-\mu\Big)\\ \end{align*}$$

converges in distribution to the standard normal random variable as $n\to \infty$, that is

$$\lim_{n\to \infty}P(Z_n\leq x)=\Phi(x),\hspace{15px}\forall x\in \mathbb{R},$$

where $\Phi(x)$ is the standard normal CDF.

Proof