Lp convergence of random variables

p次平均收敛

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

Other names / variants: mean-square convergence when p=2 · 均方收敛（p=2）

Translation evidence: Literature usage

Meaning in context

For $p\ge1$, random variables $X_n,X$ on one probability space with finite $p$th moments converge in $L^p$ when $E|X_n-X|^p\to0$.

p≥1时，具有有限p阶矩的同空间随机变量，其误差绝对值的p次方期望趋于0。

Example

A deterministic sequence $X_n=1/n$ converges to zero in every $L^p$, $p\ge1$.

Usage and distinctions

It controls error size as well as frequency; almost sure convergence alone does not suffice.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. 李东风 — 应用随机过程（北京大学）§1.4 定义1.14(3)，p阶误差积分公式 · 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(3)：p次平均收敛及均方收敛 · Checked 2026-10-14

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.