On maximal agreement couplings

TL;DR

This paper introduces the concept of maximal agreement couplings for stochastic processes, proving their existence and providing a method to bound the disagreement time independently of one process.

Contribution

It establishes the existence of maximal agreement couplings and presents a way to construct a lower bound on disagreement time independent of one process.

Findings

Maximal agreement couplings always exist.

A lower bound on disagreement time can be constructed independently of one process.

The approach applies to general stochastic processes.

Abstract

We call a coupling of two stochastic processes which maximizes the time until the first disagreement a maximal agreement coupling. We show that such a coupling always exists. Furthermore, it is possible to construct a lower bound on the disagreement time which is independent of one of the two processes.