Editing Central limit theorem (section)

===Classical CLT===
Let <math>\{X_1, \ldots, X_n}\</math> be a sequence of [[Independent and identically distributed random variables|i.i.d. random variables]] having a distribution with [[expected value]] given by <math>\mu</math> and finite [[variance]] given by <math>\sigma^2.</math> Suppose we are interested in the [[sample mean|sample average]]

<math display="block">\bar{X}_n \equiv \frac{X_1 + \cdots + X_n}{n}.</math>

By the [[law of large numbers]], the sample average [[Almost sure convergence|converges almost surely]] (and therefore also [[Convergence in probability|converges in probability]]) to the expected value <math>\mu</math> as <math>n\to\infty.</math>

The classical central limit theorem describes the size and the distributional form of the {{linktext|stochastic}} fluctuations around the deterministic number <math>\mu</math> during this convergence. More precisely, it states that as <math>n</math> gets larger, the distribution of the normalized mean <math>\sqrt{n}(\bar{X}_n - \mu)</math>, i.e. the difference between the sample average <math>\bar{X}_n</math> and its limit <math>\mu,</math> scaled by the factor <math>\sqrt{n}</math>, approaches the [[normal distribution]] with mean <math>0</math> and variance <math>\sigma^2.</math> For large enough <math>n,</math> the distribution of <math>\bar{X}_n</math> gets arbitrarily close to the normal distribution with mean <math>\mu</math> and variance <math>\sigma^2/n.</math>

The usefulness of the theorem is that the distribution of <math>\sqrt{n}(\bar{X}_n - \mu)</math> approaches normality regardless of the shape of the distribution of the individual <math>X_i.</math> Formally, the theorem can be stated as follows:

{{math theorem | name = Lindeberg–Lévy CLT | math_statement =
Suppose <math> X_1, X_2, X_3 \ldots</math> is a sequence of [[independent and identically distributed|i.i.d.]] random variables with <math>\operatorname E[X_i] = \mu</math> and <math>\operatorname{Var}[X_i] = \sigma^2 < \infty.</math> Then, as <math>n</math> approaches infinity, the random variables <math>\sqrt{n}(\bar{X}_n - \mu)</math> [[convergence in distribution|converge in distribution]] to a [[normal distribution|normal]] <math>\mathcal{N}(0, \sigma^2)</math>:{{sfnp|Billingsley|1995|p=357}}

<math display="block">\sqrt{n}\left(\bar{X}_n - \mu\right) \mathrel{\overset{d}{\longrightarrow}} \mathcal{N}\left(0,\sigma^2\right) .</math>}}

In the case <math>\sigma > 0,</math> convergence in distribution means that the [[cumulative distribution function]]s of <math>\sqrt{n}(\bar{X}_n - \mu)</math> converge pointwise to the cdf of the <math>\mathcal{N}(0, \sigma^2)</math> distribution: for every real number <math>z,</math>

<math display="block">\lim_{n\to\infty} \mathbb{P}\left[\sqrt{n}(\bar{X}_n-\mu) \le z\right] = \lim_{n\to\infty} \mathbb{P}\left[\frac{\sqrt{n}(\bar{X}_n-\mu)}{\sigma } \le \frac{z}{\sigma}\right]= 
\Phi\left(\frac{z}{\sigma}\right) ,</math>

where <math>\Phi(z)</math> is the standard normal cdf evaluated at <math>z.</math> The convergence is uniform in <math>z</math> in the sense that

<math display="block">\lim_{n\to\infty}\;\sup_{z\in\R}\;\left|\mathbb{P}\left[\sqrt{n}(\bar{X}_n-\mu) \le z\right] - \Phi\left(\frac{z}{\sigma}\right)\right| = 0~,</math>

where <math>\sup</math> denotes the least upper bound (or [[supremum]]) of the set.{{sfnp|Bauer|2001|loc=Theorem 30.13|p=199}}