submartingale

下鞅

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Translation evidence: Literature usage

Meaning in context

An integrable adapted discrete-time process $X_n$ with $E[X_{n+1}\mid\mathcal F_n]\ge X_n$ almost surely for every $n$.

各时刻可积且适应指定信息流，下一时刻的条件期望不小于当前值的离散过程。

Example

For a square-integrable martingale $M_n$, the process $M_n^2$ is a submartingale.

Usage and distinctions

The Chinese 下鞅 corresponds to conditional increase; this record requires full integrability, unlike broader one-sided conventions.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Rick Durrett — Probability: Theory and Examples, fifth edition§4.2 Martingales, Almost Sure Convergence, submartingale definition and Theorem 4.2.6 · Checked 2026-10-14

Translation sources

1. 李东风 — 应用随机过程，第6章 鞅（北京大学）§6.1.1 定义，定义6.2(6.3) · Checked 2026-10-14

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