law of large numbers · Jiayu Chen

大数定律

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

Other names / variants: LLN

Translation evidence: Literature usage

Meaning in context

In the iid integrable setting, the sample averages $n^{-1}\sum_{j=1}^n X_j$ converge to $E[X_1]$ almost surely (strong law), and therefore in probability (weak law), assuming real iid $X_j$ and $E|X_1|<\infty$ .

对独立同分布且绝对一阶矩有限的实随机变量，样本均值几乎必然收敛于共同数学期望（强大数定律），从而也依概率收敛（弱大数定律）。

Example

For iid Bernoulli observations with parameter $p$ , the observed success fraction converges almost surely to $p$ .

Usage and distinctions

This is the iid finite-mean formulation of a family of laws. Finite variance is not required here; the classical normal central limit theorem additionally assumes finite positive variance.

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Definition sources

Rick Durrett — Probability: Theory and Examples, fifth edition§2.4, Theorem 2.4.1 (pairwise independent identically distributed integrable variables; iid specialization) · Checked 2026-10-15

Translation sources

李东风 — 概率统计课件（北京大学）§4.6.1, printed pp.150–151, Theorem 6.1 headed 大数定律 and subsequent 强大数定律 discussion; name evidence from a finite-variance presentation · Checked 2026-10-15

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