Segregating Markov chains

TL;DR

This paper investigates a unique class of finite Markov chains where coupled copies meet almost surely but maintain a significant total variation distance, revealing a maximum bound of 1/2.

Contribution

It introduces and analyzes a peculiar type of Markov chain where coupling does not imply convergence in distribution, establishing a maximum total variation distance of 1/2.

Findings

Coupled chains can meet almost surely without their distributions converging.

Maximum total variation distance in this context is 1/2.

The phenomenon challenges traditional assumptions about coupling and convergence.

Abstract

Dealing with finite Markov chains in discrete time, the focus often lies on convergence behavior and one tries to make different copies of the chain meet as fast as possible and then stick together. There is, however, a very peculiar kind of discrete finite Markov chain, for which two copies started in different states can be coupled to meet almost surely in finite time, yet their distributions keep a total variation distance bounded away from 0, even in the limit as time goes off to infinity. We show that the supremum of total variation distance kept in this context is $\frac12$.