Random Walks, Directed Cycles, and Markov Chains

IF 0.4 4区数学 Q4 MATHEMATICS

American Mathematical Monthly Pub Date : 2022-12-12 DOI:10.1080/00029890.2022.2144088

K. Gingell, F. Mendivil

引用次数: 0

Abstract

Abstract A Markov chain is a random process which iteratively travels around in its state space with each transition only depending on the current position and not on the past. When the state space is discrete, we can think of a Markov chain as a special type of random walk on a directed graph. Although a Markov chain normally never settles down but keeps moving around, it does usually have a well-defined limiting behavior in a statistical sense. A given finite directed graph can potentially support many different random walks or Markov chains and each one could have one or more invariant (stationary) distributions. In this paper we explore the question of characterizing the set of all possible invariant distributions. The answer turns out to be quite simple and very natural and involves the cycles on the graph.

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随机漫步，有向环和马尔可夫链

摘要马尔可夫链是一个随机过程，它在状态空间中迭代移动，每次转移只依赖于当前位置而不依赖于过去位置。当状态空间是离散的时候，我们可以把马尔可夫链看作是有向图上的一种特殊类型的随机漫步。虽然马尔可夫链通常不会稳定下来，而是不断移动，但从统计意义上讲，它通常具有定义良好的极限行为。给定的有限有向图可以潜在地支持许多不同的随机漫步或马尔可夫链，每个随机漫步或马尔可夫链可以有一个或多个不变(平稳)分布。在本文中，我们探讨了刻画所有可能不变分布的集合的问题。答案很简单，很自然，涉及到图上的循环。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

American Mathematical Monthly Mathematics-General Mathematics

CiteScore

0.80

自引率

20.00%

发文量

127

审稿时长

6-12 weeks

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