# Sequences and Limit Theorems

Sequences :: Convergence :: Limit :: Central Limit

## Central Limit Theorems

**Theorem 3**

**Central Limit Theorem**Suppose is a sequence of i.i.d. random variables with mean and variance . Then

*where*

*That is,*

The main point: Sums of i.i.d. random variables tend to look

**Gaussian**.

To work our way up to this, here are a couple of lemmas:

**Lemma 2**

*Suppose is a sequence of r.v.s with characteristic functions . If there exists a r.v. with ch.f. such that*

*for all then*

**Lemma 3**

*Suppose is a r.v. with . Then has the expansion*

*where .*

Summarizing, if
ahas zero mean and variance 1,