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

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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

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

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

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

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

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This means that
image6053image6054

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

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

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From which it follows that
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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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