Central Limit Theorem 的證明?

Classical central limit theorem
The theorem most often called the central limit theorem is the following. Let X1, X2, X3, ... be a sequence of random variables which are defined on the same probability space, share the same probability distribution D and are independent. Assume that both the expected value μ and the standard deviation σ of D exist and are finite.
Consider the sum Sn = X1 + ... + Xn. Then the expected value of Sn is nμ and its standard deviation is σ * n½. Furthermore, informally speaking, the distribution of Sn approaches the normal distribution N(nμ,σ2n) as n approaches ∞.
In order to clarify the word "approaches" in the last sentence, we standardize Sn by setting


圖片參考：http://upload.wikimedia.org/math/c/0/b/c0be5ec3af28935d8da580d1eb5d7bfd.png

Then the distribution of Zn converges towards the standard normal distribution N(0,1) as n approaches ∞ (this is convergence in distribution). This means: if Φ(z) is the cumulative distribution function of N(0,1), then for every real number z, we have


圖片參考：http://upload.wikimedia.org/math/7/8/5/785183e0b3c75998b43395413bf25776.png

or, equivalently,


圖片參考：http://upload.wikimedia.org/math/9/e/6/9e6dce464d5faafce318ac79a915eda3.png

where


圖片參考：http://upload.wikimedia.org/math/6/b/2/6b21c75eac21a59c111dcc285a25435e.png

is the sample mean.

Proof of the central limit theorem
For a theorem of such fundamental importance to statistics and applied probability, the central limit theorem has a remarkably simple proof using characteristic functions. It is similar to the proof of a (weak) law of large numbers. For any random variable, Y, with zero mean and unit variance (var(Y) = 1), the characteristic function of Y is, by Taylor's theorem,


圖片參考：http://upload.wikimedia.org/math/2/c/b/2cbfb53bb40b7b7229b8907bd8973a65.png

where o (t2 ) is "little o notation" for some function of t that goes to zero more rapidly than t2. Letting Yi be (Xi − μ)/σ, the standardised value of Xi, it is easy to see that the standardised mean of the observations X1, X2, ..., Xn is just


圖片參考：http://upload.wikimedia.org/math/e/0/a/e0a81d1b173ef5dd9fc7a76c505a19d2.png

By simple properties of characteristic functions, the characteristic function of Zn is


圖片參考：http://upload.wikimedia.org/math/5/1/4/5147da59182098ae4ee80d225aa23ed5.png

But, this limit is just the characteristic function of a standard normal distribution, N(0,1), and the central limit theorem follows from the Lévy continuity theorem, which confirms that the convergence of characteristic functions implies convergence in distribution.