Central limits theorem

The central limits theorem states that as the sample size $𝑛$ increases, the sample mean $―― 𝑋 𝑛$ converges in distribution to a normal distribution, regardless of the underlying distribution of $𝑋$ . #m/thm/stat That is,

―― 𝑋 𝑛 ⇝ N (𝜇 𝑋, 𝜎 2 𝑋 𝑛)

or equivalently

―― 𝑋 𝑛 - 𝜇 𝑋 𝜎 / \sqrt 𝑛 ⇝ N (0, 1)

as $𝑛 \to \infty$ . In the case where $𝑋$ itself is normally distributed, $―― 𝑋 𝑛$ is already normal for all $𝑛$ . Otherwise, $𝑛 = 30$ is generally taken as a good guide.

Proof¹

Consider a set of independent, similarly distributed random variables $𝑋 1 \sim 𝑋 2 \sim \dots \sim 𝑋 𝑛$ with expected value $𝜇$ and probability density function $𝑤 (𝑥)$ . It is useful to introduce the Random function

𝑌 = \sum 𝑛 𝑖 = 1 (𝑋 𝑖 - 𝜇) \sqrt 𝑛 = \sum 𝑛 𝑖 = 1 𝑋 𝑖 \sqrt 𝑛 - \sqrt 𝑛 𝜇

which by Distribution has distribution

𝑤 𝑌 (𝑧) = ⟨ 𝛿 (𝑌 (⃗ 𝐗) - 𝑦) ⟩ = \int ℝ 𝑛 (𝑛 \prod 𝑗 = 1 𝑤 (𝑥 𝑗) 𝑑 𝑥 𝑗) 𝛿 (𝑦 - 1 \sqrt 𝑛 𝑛 \sum 𝑗 = 1 (𝑥 𝑗 - 𝜇)) = 1 2 𝜋 \int ℝ 𝑛 (𝑛 \prod 𝑗 = 1 𝑤 (𝑥 𝑗) 𝑑 𝑥 𝑗) \int \infty - \infty 𝑑 𝑘 e x p (𝑖 𝑘 (𝑦 - 1 \sqrt 𝑛 𝑛 \sum 𝑗 = 1 (𝑥 𝑗 - 𝜇))) = 1 2 𝜋 \int \infty - \infty 𝑑 𝑘 𝑒 𝑖 𝑘 𝑦 \int ℝ 𝑛 𝑛 \prod 𝑗 = 1 𝑑 𝑥 𝑗 𝑤 (𝑥 𝑗) e x p (- 𝑖 𝑘 \sqrt 𝑛 (𝑥 𝑗 - 𝜇)) = 1 2 𝜋 \int \infty - \infty 𝑑 𝑘 𝑒 𝑖 𝑘 𝑦 + 𝑖 𝑘 \sqrt 𝑛 𝜇 (\int \infty - \infty 𝑑 𝑥 𝑤 (𝑥) e x p (- 𝑖 𝑘 𝑥)) 𝑛 = 1 2 𝜋 \int \infty - \infty 𝑑 𝑘 𝑒 𝑖 𝑘 𝑦 + 𝑖 𝑘 \sqrt 𝑛 𝜇 𝜒 (𝑘 \sqrt 𝑛) 𝑛

as required.

#state/tidy | #SemBr | #lang/en

2006, Statistische Mechanik, p. 8 ↩

Central limits theorem

Footnotes