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Central limit theorem

Central limit theorems are a set of weak-convergence results in probability theory. Intuitively, they all express the fact that any sum of many independent identically distributed random variables is approximately normally distributed. These results explain the ubiquity of the normal distribution.

The most important and famous result is simply called The Central Limit Theorem; it is concerned with independent variables with identical distribution whose expected value and variance are finite. Several generalizations exist which do not require identical distribution but incorporate some condition which guarantees that none of the variables exert a much larger influence than the others. Two such conditions are the Lindeberg condition and the Lyapunov condition. Other generalizations even allow some "weak" dependence of the random variables.

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"The" central limit theorem

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 normalize Sn by setting

<math>Z_n = \frac{S_n - n \mu}{\sigma \sqrt{n}}</math>

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

<math>\lim_{n \to \infty} \mbox{Pr}(Z_n \le z) = \Phi(z),</math>
or, equivalently,
<math>\lim_{n\rightarrow\infty}\mbox{Pr}\left(\frac{\overline{X}_n-\mu}{\sigma/\sqrt{n}}\leq z\right)=\Phi(z)</math>
is the "sample mean".

If the third central moment E((X1-μ)3) exists and is finite, then the above convergence is uniform and the speed of convergence is at least on the order of 1/n½ (see Berry-Esséen theorem[?]).

picture of a distribution being "smoothed out" by summation would be nice

An equivalent formulation of this limit theorem starts with An = (X1 + ... + Xn) / n which can be interpreted as the mean of a random sample of size n. The expected value of An is μ and the standard deviation is σ / n½. If we normalize An by setting Zn = (An - μ) / (σ / n½), we obtain the same variable Zn as above, and it approaches a standard normal distribution.

Note the following "paradox": by adding many independent identically distributed positive variables, one gets approximately a normal distribution. But for every normally distributed variable, the probability that it is negative is non-zero! How is it possible to get negative numbers from adding only positives? The key lies in the word "approximately". The sum of positive variables is of course always positive, but it is very well approximated by a normal variable (which indeed has a very tiny probability of being negative).

More precisely: the fact that, in this case, for every n there is a z such that Pr(Znz) = 0 does not contradict that for every z we have limn→∞ Pr(Znz) > 0, because in the first case z may depend on n and in the second case n is increased for a fixed z.

Lyapunov condition

Let Xn be a sequence of independent random variables defined on the same probability space. Assume that Xn has finite expected value μn and finite standard deviation σn. We define

<math>s_n^2 = \sum_{i = 1}^n \sigma_i^2</math>

Assume that the third central moments

<math>r_n^3 = \mbox{E}\left({\left| X_n - \mu_n \right|}^3 \right)</math>

are finite for every n, and that

<math>\lim_{n \to \infty} \frac{r_n}{s_n} = 0</math>

(This is the Lyapunov condition.) We again consider the sum Sn=X1+...+Xn. The expected value of Sn is mn = ∑i=1..nμi and its standard deviation is sn. If we normalize Sn by setting

<math>Z_n = \frac{S_n - m_n}{s_n}</math>

then the distribution of Zn converges towards the standard normal distribution N(0,1) as above.

Lindeberg condition

In the same setting and with the same notation as above, we can replace the Lyapunov condition with the following weaker one: for every ε > 0

  \lim_{n \to \infty} \sum_{i = 1}^{n} \mbox{E}\left(
    \frac{(X_i - \mu_i)^2}{s_n^2}
    \left| X_i - \mu_i \right| > \epsilon s_n
  \right) = 0

(where E( U : V > c) denotes the conditional expected value: the expected value of U given that V > c.) Then the distribution of the normalized sum Zn converges towards the standard normal distribution N(0,1).

Non-independent case

There are some theorems which treat the case of sums of non-independent variables, for instance the m-dependent central limit theorem, the martingale central limit theorem and the central limit theorem for mixing processes.

track these down

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