Editing Central limit theorem (section)

==Dependent processes==

===CLT under weak dependence===
A useful generalization of a sequence of independent, identically distributed random variables is a [[Mixing (mathematics)|mixing]] random process in discrete time; "mixing" means, roughly, that random variables temporally far apart from one another are nearly independent. Several kinds of mixing are used in ergodic theory and probability theory. See especially [[Mixing (mathematics)#Mixing in stochastic processes|strong mixing]] (also called α-mixing) defined by <math display="inline">\alpha(n) \to 0</math> where <math display="inline">\alpha(n)</math> is so-called [[Mixing (mathematics)#Mixing in stochastic processes|strong mixing coefficient]].

A simplified formulation of the central limit theorem under strong mixing is:{{sfnp|Billingsley|1995|loc=Theorem 27.4}}

{{math theorem | math_statement = Suppose that <math display="inline">\{X_1, \ldots, X_n, \ldots\}</math> is stationary and <math>\alpha</math>-mixing with <math display="inline">\alpha_n = O\left(n^{-5}\right) </math> and that <math display="inline">\operatorname E[X_n] = 0</math> and {{nowrap|<math display="inline">\operatorname E[X_n^{12}] < \infty</math>.}} Denote {{nowrap|<math display="inline">S_n = X_1 + \cdots + X_n</math>,}} then the limit

<math display="block"> \sigma^2 = \lim_{n\rightarrow\infty} \frac{\operatorname E\left(S_n^2\right)}{n} </math>

exists, and if <math display="inline">\sigma \ne 0</math> then <math display="inline">\frac{S_n}{\sigma\sqrt{n}}</math> converges in distribution to <math display="inline"> \mathcal{N}(0, 1)</math>.}}

In fact,

<math display="block">\sigma^2  = \operatorname E\left(X_1^2\right) + 2 \sum_{k=1}^{\infty} \operatorname E\left(X_1 X_{1+k}\right),</math>

where the series converges absolutely.

The assumption <math display="inline">\sigma \ne 0</math> cannot be omitted, since the asymptotic normality fails for <math display="inline">X_n = Y_n - Y_{n-1}</math> where <math display="inline">Y_n</math> are another [[stationary sequence]].

There is a stronger version of the theorem:{{sfnp|Durrett|2004|loc=Sect. 7.7(c), Theorem 7.8}} the assumption <math display="inline">\operatorname E\left[X_n^{12}\right] < \infty</math> is replaced with {{nowrap|<math display="inline">\operatorname E\left[{\left|X_n\right|}^{2+\delta}\right] < \infty</math>,}} and the assumption <math display="inline">\alpha_n = O\left(n^{-5}\right) </math> is replaced with

<math display="block">\sum_n \alpha_n^{\frac\delta{2(2+\delta)}} < \infty.</math>

Existence of such <math display="inline">\delta > 0</math> ensures the conclusion. For encyclopedic treatment of limit theorems under mixing conditions see {{harv|Bradley|2007}}.

===Martingale difference CLT===
{{Main|Martingale central limit theorem}}
{{math theorem | math_statement = Let a [[Martingale (probability theory)|martingale]] <math display="inline">M_n</math> satisfy
* <math> \frac1n \sum_{k=1}^n \operatorname E\left[\left(M_k-M_{k-1}\right)^2 \mid M_1,\dots,M_{k-1}\right] \to 1 </math>  in probability as {{math|''n'' → ∞}},
* for every {{math|''ε'' > 0}}, <math> \frac1n \sum_{k=1}^n{\operatorname E\left[\left(M_k-M_{k-1} \right)^2\mathbf{1}\left[|M_k-M_{k-1}|>\varepsilon\sqrt{n}\right]\right]} \to 0 </math>  as {{math|''n'' → ∞}},

then <math display="inline">\frac{M_n}{\sqrt{n}}</math> converges in distribution to <math display="inline">\mathcal{N}(0, 1)</math> as <math display="inline">n \to \infty</math>.{{sfnp|Durrett|2004|loc=Sect. 7.7, Theorem 7.4}}{{sfnp|Billingsley|1995 |loc=Theorem 35.12}}}}