论文标题
从过去的耦合到无效的马尔可夫链
Coupling from the Past for the Null Recurrent Markov Chain
论文作者
论文摘要
可数的状态空间马尔可夫链的多布林图描述了马尔可夫动力学从所有可能的初始条件开始的关节路线演变，当它们同时达到状态空间的同一点时，两条路径合并在一起。它的桥乌布林子图仅包含从状态空间的标记点开始的路径。在不可还原，多个阳性复发的情况下，已知以下结果：桥乌布林图是一棵无限型的无限树。此外，它包含一条单一的双足路径，该路径允许一个人建立马尔可夫链的固定状态的完美样本。本文集中在无效的情况下。结果表明，当再次假设不可约性和多个神经性时，桥梁的桥图是无限的树，也可以是由可数的无限树组成的森林。在第一种情况下，所讨论的无限树具有单一端，通常不可测量，但始终是局部不可测量的。这些关键特性用于研究该桥宝林树上几种测量值随机动力学的固定状态。最重要的是禁忌随机动力学，它作为稳态作为一种随机度量，平均度量等于马尔可夫链的不变度，而潜在的随机动力学是经典潜在测度的随机扩展，其平均度量在状态空间的每个点上等于无穷大。
The Doeblin Graph of a countable state space Markov chain describes the joint pathwise evolutions of the Markov dynamics starting from all possible initial conditions, with two paths coalescing when they reach the same point of the state space at the same time. Its Bridge Doeblin subgraph only contains the paths starting from a tagged point of the state space at all possible times. In the irreducible, aperiodic, and positive recurrent case, the following results are known: the Bridge Doeblin Graph is an infinite tree that is unimodularizable. Moreover, it contains a single bi-infinite path, which allows one to build a perfect sample of the stationary state of the Markov chain. The present paper is focused on the null recurrent case. It is shown that when assuming irreducibility and aperiodicity again, the Bridge Doeblin Graph is either an infinite tree or a forest made of a countable collection of infinite trees. In the first case, the infinite tree in question has a single end, is not unimodularizable in general, but is always locally unimodular. These key properties are used to study the stationary regime of several measure-valued random dynamics on this Bridge Doeblin Tree. The most important ones are the taboo random dynamics, which admits as steady state a random measure with mean measure equal to the invariant measure of the Markov chain, and the potential random dynamics which is a random extension of the classical potential measure, with a mean measure equal to infinity at every point of the state space.
