Markov chain

马尔可夫链

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

Translation evidence: Literature usage

Meaning in context

In the finite-state time-homogeneous discrete-time setting, a process $X_n$ with initial law $\alpha$ and row-stochastic matrix $P=(p_{ij})$ satisfies $P(X_{n+1}=j\mid\sigma(X_0,\ldots,X_n))=p_{X_nj}$ almost surely for every $n\ge0$ and state $j$.

在有限状态、离散时间、时间齐次的范围内，给定初始分布和各行和为1的转移矩阵后，下一步状态在全部既往观测条件下的概率只由当前状态及相应矩阵行确定。

Example

For $P=\begin{pmatrix}0.8&0.2\\0.4&0.6\end{pmatrix}$, starting from state $0$ gives probability $0.28$ of state $1$ after two steps.

Usage and distinctions

The Markov property is conditional prediction from the present, not independence of successive states. Time homogeneity is a further restriction; this record uses it and finite states explicitly.

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Definition sources

1. University of Cambridge — Markov Chains, course notes hosted by Richard Weber§1.2, Definition 1.2 and row-stochastic matrix convention, specialized from countable to finite states · Checked 2026-10-15

Translation sources

1. 陈大岳 — 应用随机过程课程（北京大学）《应用随机过程》2008年秋，主要内容：第二章 马尔可夫链，items 1–3; course-outline name evidence only · Checked 2026-10-15

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