structural:expansion, information:extraction, stylistic:tone
CLT under weak dependence

A useful generalization of a sequence of independent, identically distributed random variables is a 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 strong mixing (also called α-mixing) defined by α (n) → 0 {\textstyle \alpha (n)\to 0} where α (n) {\textstyle \alpha (n)} is so-called strong mixing coefficient.

A simplified formulation of the central limit theorem under strong mixing is:

Theorem — Suppose that { X 1, …, X n, … } {\textstyle \{X_{1},\ldots,X_{n},\ldots \}} is stationary and α {\displaystyle \alpha } -mixing with α n = O (n − 5) {\textstyle \alpha _{n}=O\left(n^{-5}\right)} and that E = 0 {\textstyle \operatorname {E} =0} and E < ∞ {\textstyle \operatorname {E} <\infty }. Denote S n = X 1 + ⋯ + X n {\textstyle S_{n}=X_{1}+\cdots +X_{n}}, then the limit σ 2 = lim n → ∞ E (S n 2) n {\displaystyle \sigma ^{2}=\lim _{n\rightarrow \infty }{\frac {\operatorname {E} \left(S_{n}^{2}\right)}{n}}} exists, and if σ ≠ 0 {\textstyle \sigma \neq 0} then S n σ n {\textstyle {\frac {S_{n}}{\sigma {\sqrt {n}}}}} converges in distribution to N (0, 1) {\textstyle {\mathcal {N}}(0,1)}.

In fact, σ 2 = E (X 1 2) + 2 ∑ k = 1 ∞ E (X 1 X 1 + k), {\displaystyle \sigma ^{2}=\operatorname {E} \left(X_{1}^{2}\right)+2\sum _{k=1}^{\infty }\operatorname {E} \left(X_{1}X_{1+k}\right),} where the series converges absolutely.

The assumption σ ≠ 0 {\textstyle \sigma \neq 0} cannot be omitted, since the asymptotic normality fails for X n = Y n − Y n − 1 {\textstyle X_{n}=Y_{n}-Y_{n-1}} where Y n {\textstyle Y_{n}} are another stationary sequence.

There is a stronger version of the theorem: the assumption E < ∞ {\textstyle \operatorname {E} \left<\infty } is replaced with E < ∞ {\textstyle \operatorname {E} \left<\infty }, and the assumption α n = O (n − 5) {\textstyle \alpha _{n}=O\left(n^{-5}\right)} is replaced with ∑ n α n δ 2 (2 + δ) < ∞. {\displaystyle \sum _{n}\alpha _{n}^{\frac {\delta }{2(2+\delta)}}<\infty.}

Existence of such δ > 0 {\textstyle \delta >0} ensures the conclusion. For encyclopedic treatment of limit theorems under mixing conditions see (Bradley 2007).