07 - Central Limit Theorem

Central Limit Theorem

Let $X_1,X_2,\cdots$ be i.i.d random variables with mean $\mu$ and variance ${\sigma }^2\in (0,\infty )$ and

$$S_n=X_1+X_2+\cdots +X_n$$

Then

$$\frac{S_n-n\mu }{\sigma \sqrt{n}}\stackrel{d}{\to }N(0,1)\text{ as }n\to \infty$$

Properties of CLT

1. Distribution of $S_n$ concentrates around $n\mu$

2. Fluctuations of $S_n$ around $n\mu$ are of order $\sqrt{n}$

3. Asymptotic Distribution of these fluctuations are normal

Alternate Forms of CLT

$$\mathbb{P}\left(a\le \frac{S_n-n\mu }{\sigma \sqrt{n}}\le b\right)\to \mathbb{P}(a\le Z\le b)\text{ as }n\to \infty \text{ for }Z\sim N(0,1)\forall a\le b$$

$$\frac{S_n-n\mu }{\sqrt{n}}\stackrel{d}{\to }N(0,{\sigma }^2)\text{ as }n\to \infty$$