The Central Limit Theorem

In this chapter we start with a simple argument that “proves” the central limit theorem. Then we take the argument and make it rigorous. There are much more general statements of the central limit theorem than the statement found here [8]. The proofs of the more general versions, however, are less elementary and do not shed much new light on what is happening. Consider the random variables Xi, assumed to be IID, to have zero as their

common expected value, and to have one as their common standard deviation. Let us consider the sum:

SN = ∑N

i=1Xi√ N

.