Central limit theorem

Statement

Let $X_1, X_2, \dots$ be i.i.d. real random variables with mean $\mu$ and variance $\sigma^2$, where $0 < \sigma^2 < \infty$. Let $\bar{X}_n = (X_1 + \dots + X_n)/n$. Then $\sqrt{n} \, (\bar{X}_n - \mu) / \sigma$ converges in distribution to $\mathcal{N}(0, 1)$. That is, for each $x \in \mathbb{R}$, $$\lim_{n \to \infty} P\left(\frac{X_1 + \dots + X_n - n\mu}{\sigma \sqrt{n}} \le x\right) = \Phi(x) = \int_{-\infty}^{x} \frac{1}{\sqrt{2\pi}} e^{-u^2/2} \, du.$$

Remarks

A standard proof uses characteristic functions. With $Y_i = (X_i - \mu)/\sigma$, $\varphi_{Y_1}(t) = 1 - t^2/2 + o(t^2)$, so $\varphi_{Y_1}(t/\sqrt{n})^n \to e^{-t^2/2}$. Lévy's continuity theorem then gives the convergence in distribution. The Lindeberg and Lyapunov forms remove the condition that the $X_i$ have the same law. The Berry–Esseen theorem bounds the rate of convergence when $E[|X_1|^3] < \infty$. The finite variance is necessary. Sums of i.i.d. Cauchy variables, scaled by $1/n$, stay Cauchy.

Explanations

L1 human

The central limit theorem explains why the bell curve is everywhere. A sum of many small independent effects is approximately normal, whatever the law of each effect, as long as the variance is finite.