convergence in probability

依概率收敛

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Translation evidence: Literature usage

Meaning in context

Variables on one probability space satisfy Xn→XX_n\to X in probability when P(∣Xn−X∣>ε)→0P(|X_n-X|>\varepsilon)\to0 for every ε>0\varepsilon>0.

同一概率空间上的变量序列，其偏离目标超过任何固定正误差的概率趋于0。

Example

If P(An)=1/nP(A_n)=1/n, then 1An→01_{A_n}\to0 in probability, whether or not the events are independent.

Usage and distinctions

This does not guarantee convergence on almost every path; the bad events may keep moving.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Gregory F. Lawler — Notes on ProbabilitySection 4, opening definition of convergence in probability · Checked 2026-10-14

Translation sources

  1. 李东风 — 应用随机过程（北京大学）§1.4 收敛性，定义1.14(2) · Checked 2026-10-14
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