Optimal stopping time and halting set for total variation distance

TL;DR

This paper investigates the relationship between optimal coupling times and hitting times of specific sets, called halting sets, in Markov chains, providing new methods to analyze cutoff phenomena and convergence rates.

Contribution

It introduces the concept of halting sets for total variation convergence and compares maximal coupling times to hitting times, extending analysis to non-lazy birth-and-death chains.

Findings

Maximal coupling time can be characterized by hitting times of halting sets.

Halting sets are identified for symmetric birth-and-death chains.

New methods for calculating cutoff times without the lazy hypothesis are proposed.

Abstract

An aperiodic and irreducible Markov chain on a finite state space converges to its stationary distribution. When convergence to equilibrium is measured by total variation distance, there exists an optimal coupling and a maximal coupling time. In this article, the maximal coupling time is compared to the hitting time of a specific state or set. Such sets, named halting sets, are studied in the case of symmetric birth-and-death chains and in some other examples. Some applications to the cutoff phenomenon are given. These results yield new methods to calculate cutoff times for some monotone birth-and death chains without the lazy hypothesis .