Central Limit Theorem – Advanced

Central Limit Theorem: If x has a distribution with mean μ and standard deviation σ then for n sufficiently large, the variable

has a distribution that is approximately the standard normal distribution.

Proof: Using Properties 3 and 4 of General Properties of Distributions, and the fact that all the x_i are independent with the same distribution, we have

Taking the natural log of both sides of the equation, we get

As we can see from the proof of Property 2 of General Properties of Distributions,

Hence

If we set z = all the terms on the right after 1 +, we have

Thus for sufficiently large n, |z| < 1. It then follows that

This means that

where q consists of all the terms in θ^k in the double expansion with k ≥ 3. Note too that the only function of n that each of these terms has in common is of the form n^{– k ⁄ 2}. Rearranging the terms, we have

Since  and , we conclude that

But as n → ∞ we see that nqθ³ → 0. Thus

from which it follows that

By Property 3 of Normal Distribution, a random variable with standard normal distribution has the same moment generating function, and so the result follows from Corollary 1 of General Properties of Distributions.

References

Soch, J. (2020) Proof: Moment-generating function of the normal distribution. The book of statistical proofs
https://statproofbook.github.io/P/norm-mgf.html

Hoel, P. G. (1962) Introduction to mathematical statistics. Wiley