Central Limit Theorem

Lindeberg-Levy CLT

Suppose $X_1,X_2,\dots$ is a sequence of i.i.d. random variables with $\mathrm{E}[X_i]=\mu$ and $\mathrm{Var}(X_i)={\sigma }^2<\infty$. Then, as $n\to \infty$, the random variable $\sqrt{n}({\bar{X}}_n-\mu )$ converge in distribution to a normal $\mathcal{N}(0,{\sigma }^2)$:

$$\sqrt{n}\left({\bar{X}}_n-\mu \right)\stackrel{d}{\to }\mathcal{N}(0,{\sigma }^2)$$

or

$$\frac{\left({\bar{X}}_n-\mu \right)}{\mathrm{SE}}\stackrel{d}{\to }\mathcal{N}(0,1)$$

where

$${\bar{X}}_n=\frac{1}{n}\left(X_1+X_2+\cdots +X_n\right)$$

is the sample mean, and $\mathrm{SE}=\frac{\sigma }{\sqrt{n}}$ is the standard error.