central limit theorem

中心极限定理

Probability · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: CLT

Translation evidence: Literature usage

Meaning in context

The classical iid central limit theorem states that for real iid $X_j$ with mean $\mu$ and variance $0<\sigma^2<\infty$, $(\sum_{j=1}^n X_j-n\mu)/(\sigma\sqrt n)$ converges in distribution to $N(0,1)$.

经典独立同分布中心极限定理断言：实随机变量列独立同分布、均值为μ且方差σ²有限并严格为正时，和减去nμ再除以σ√n后依分布收敛于标准正态分布。

Example

For independent Bernoulli trials with fixed $0<p<1$, $(S_n-np)/\sqrt{np(1-p)}$ converges in distribution to $N(0,1)$.

Usage and distinctions

The theorem concerns centered, scaled sums, not the original observations becoming normal. This classical iid version does not cover arbitrary dependence or infinite variance.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Gregory F. Lawler — Notes on ProbabilitySection 5, Proposition 5.4 and Theorem 5.5; standardization requires positive variance · Checked 2026-10-15

Translation sources

1. 李东风 — 概率统计课件（北京大学）§4.6.2, printed p.153, Theorem 6.2 headed 中心极限定理 · Checked 2026-10-15

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