Theorem

Let X1,X2,...X_1, X_2,... be i.i.d. random variables with expected value 𝔼[Xi]=η\mathbb{E}[X_i] = \eta and finite variance Var⁡(Xi)=σ2\operatorname{Var}(X_i) = \sigma^2, and define YnY_n as

Yn=X1+...+XnnY_n = \frac{X_1 + ... + X_n}{n}

then, n(Yn−η)→X\sqrt{n}(Y_n - \eta) \to X in distribution (converges in distribution to XX), where X∼𝒩(0,σ2)X \sim \mathcal{N}(0,\sigma^2) (Gaussian distribution)

(n(Yn−η)→d𝒩(0,σ2)\sqrt{n}(Y_n - \eta) \overset{d}{\to} \mathcal{N}(0,\sigma^2))

CLT - Informal

Any sum of mutually independent, (identically distributed) random variables X1,…,XkX_1,…,X_k with mean μ\mu and finite variance σ2\sigma^2 converges to a Gaussian random variable with mean k⋅μk \cdot \mu and variance k⋅σ2k \cdot \sigma^2, as k→∞k \rightarrow \infty.

S=∑i=1kXi⟹𝒩(k⋅μ,k⋅σ2)S = \sum_{i=1}^k X_i \implies \mathcal{N}(k \cdot \mu, k \cdot \sigma^2)

See also

References:

  1. https://math.mit.edu/~sheffield/2018600/Lecture22.pdf
  2. https://www.probabilitycourse.com/chapter7/7_1_2_central_limit_theorem.php
  3. https://people.math.harvard.edu/~knill/teaching/math19b_2011/handouts/lecture32.pdf
  4. https://en.wikipedia.org/wiki/Central_limit_theorem#Classical_CLT