Editing Central limit theorem (section)

===Proof of classical CLT===
The central limit theorem has a proof using [[characteristic function (probability theory)|characteristic functions]].<ref>{{cite book|url=https://jhupbooks.press.jhu.edu/content/introduction-stochastic-processes-physics|title=An Introduction to Stochastic Processes in Physics|publisher=Johns Hopkins University Press|year=2003 |doi=10.56021/9780801868665 |access-date=2016-08-11 |last1=Lemons |first1=Don |isbn=9780801876387 }}</ref> It is similar to the proof of the (weak) [[Proof of the law of large numbers|law of large numbers]].

Assume <math display="inline">\{X_1, \ldots, X_n, \ldots \}</math> are independent and identically distributed random variables, each with mean <math display="inline">\mu</math> and finite variance {{nowrap|<math display="inline">\sigma^2</math>.}} The sum <math display="inline">X_1 + \cdots + X_n</math> has [[Linearity of expectation|mean]] <math display="inline">n\mu</math> and [[Variance#Sum of uncorrelated variables (Bienaymé formula)|variance]] {{nowrap|<math display="inline">n\sigma^2</math>.}} Consider the random variable

<math display="block">Z_n = \frac{X_1+\cdots+X_n - n \mu}{\sqrt{n \sigma^2}} = \sum_{i=1}^n \frac{X_i - \mu}{\sqrt{n \sigma^2}} = \sum_{i=1}^n \frac{1}{\sqrt{n}} Y_i,</math>

where in the last step we defined the new random variables {{nowrap|<math display="inline">Y_i = \frac{X_i - \mu}{\sigma} </math>,}} each with zero mean and unit variance {{nowrap|(<math display="inline">\operatorname{var}(Y) = 1</math>).}} The [[Characteristic function (probability theory)|characteristic function]] of <math display="inline">Z_n</math> is given by

<math display="block">\varphi_{Z_n}\!(t) = \varphi_{\sum_{i=1}^n {\frac{1}{\sqrt{n}}Y_i}}\!(t) \ =\ \varphi_{Y_1}\!\!\left(\frac{t}{\sqrt{n}}\right) \varphi_{Y_2}\!\! \left(\frac{t}{\sqrt{n}}\right)\cdots \varphi_{Y_n}\!\! \left(\frac{t}{\sqrt{n}}\right) \ =\ \left[\varphi_{Y_1}\!\!\left(\frac{t}{\sqrt{n}}\right)\right]^n,
</math>

where in the last step we used the fact that all of the <math display="inline">Y_i</math> are identically distributed. The characteristic function of <math display="inline">Y_1</math> is, by [[Taylor's theorem]],
<math display="block">\varphi_{Y_1}\!\left(\frac{t}{\sqrt{n}}\right) = 1 - \frac{t^2}{2n} + o\!\left(\frac{t^2}{n}\right), \quad \left(\frac{t}{\sqrt{n}}\right) \to 0</math>

where <math display="inline">o(t^2 / n)</math> is "[[Little-o notation|little {{mvar|o}} notation]]" for some function of <math display="inline">t</math> that goes to zero more rapidly than {{nowrap|<math display="inline">t^2 / n</math>.}} By the limit of the [[exponential function]] {{nowrap|(<math display="inline">e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n</math>),}} the characteristic function of <math>Z_n</math> equals

<math display="block">\varphi_{Z_n}(t) = \left(1 - \frac{t^2}{2n} + o\left(\frac{t^2}{n}\right) \right)^n \rightarrow e^{-\frac{1}{2} t^2}, \quad n \to \infty.</math>

All of the higher order terms vanish in the limit {{nowrap|<math display="inline">n\to\infty</math>.}} The right hand side equals the characteristic function of a standard normal distribution <math display="inline">\mathcal{N}(0, 1)</math>, which implies through [[Lévy continuity theorem|Lévy's continuity theorem]] that the distribution of <math display="inline">Z_n</math> will approach <math display="inline">\mathcal{N}(0,1)</math> as {{nowrap|<math display="inline">n\to\infty</math>.}} Therefore, the [[sample mean|sample average]]

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

is such that 

<math display="block">\frac{\sqrt{n}}{\sigma}(\bar{X}_n - \mu) = Z_n</math>

converges to the normal distribution {{nowrap|<math display="inline">\mathcal{N}(0, 1)</math>,}} from which the central limit theorem follows.