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Central limit theorem

The classical limit theorem stating that normalized sums of i.i.d. variables converge in distribution to a normal law.

Central limit theorem (i.i.d. version): Let (Xi)i≥1(X_i)_{i\ge 1} be an of real-valued with E[X1]=μ\mathbb{E}[X_1]=\mu and Var(X1)=σ2\mathrm{Var}(X_1)=\sigma^2 where 0<σ2<∞0<\sigma^2<\infty. Define Sn=∑i=1nXiS_n=\sum_{i=1}^n X_i. Then

Sn−nμσn⇒N(0,1)as n→∞,\frac{S_n-n\mu}{\sigma\sqrt{n}} \Rightarrow \mathcal{N}(0,1)\quad\text{as }n\to\infty,

where ⇒\Rightarrow denotes convergence in distribution.

Remarks

The theorem connects and to the asymptotic of sums, and it underlies normal approximations used throughout probability and statistics.