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

Central limit theorem (i.i.d. version): Let $(X_i)_{i\ge 1}$ be an i.i.d. sequence of real-valued random variables with $\mathbb{E}[X_1]=\mu$ and $\mathrm{Var}(X_1)=\sigma^2$ where $0<\sigma^2<\infty$ . Define $S_n=\sum_{i=1}^n X_i$ . Then

\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.

The theorem connects expectation and variance to the asymptotic distribution (law) of sums, and it underlies normal approximations used throughout probability and statistics.