almost sure convergence

几乎必然收敛

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Other names / variants: a.s. convergence · convergence with probability one · 以概率1收敛

Translation evidence: Literature usage

Meaning in context

On one probability space, Xn→XX_n\to X almost surely if P({ω:lim⁡nXn(ω)=X(ω)})=1P(\{\omega:\lim_nX_n(\omega)=X(\omega)\})=1.

在同一概率空间上，存在概率为1的事件，使变量序列在其中每个样本点上收敛于目标变量。

Example

For uniform U∈(0,1)U\in(0,1), n1{U<1/n}→0n1_{\{U<1/n\}}\to0 almost surely although every expectation is 11.

Usage and distinctions

The exceptional null set is chosen for the whole sequence, not separately to assert convergence at one time.

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Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Gregory F. Lawler — Notes on ProbabilitySection 4, opening definition of almost-sure convergence · 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(1) · Checked 2026-10-14
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