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 which is approximately the standard normal distribution.

Proof: Using Properties 3 and 4 of General Properties of Distributions, and the fact that all the xi 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,


And so

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 θk in the double expansion with k ≥ 3. Note too that the only function of n that each of these terms have is of the form n– k ⁄ 2. Rearranging the terms, we have

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Since \mu'_1 = \mu and \sigma^2 = \mu'_2 - (\mu'_1)^2, 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.

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