Convergence rate bounds for iterative random functions using one-shot coupling

TL;DR

This paper introduces the One-Shot Coupling Theorem, providing a unified framework for bounding convergence rates of Markov chains without exogenous variables, and demonstrates its effectiveness on various models including high-dimensional cases.

Contribution

The paper formalizes the one-shot coupling method into a theorem and applies it to different Markov chains, showing it yields tight convergence bounds without needing drift or minorization conditions.

Findings

Theorem provides straightforward verification of convergence conditions.

Method yields tight geometric convergence rate bounds.

Applicable to high-dimensional Markov chain models.

Abstract

One-shot coupling is a method of bounding the convergence rate between two copies of a Markov chain in total variation distance, which was first introduced by Roberts and Rosenthal and generalized by Madras and Sezer. The method is divided into two parts: the contraction phase, when the chains converge in expected distance and the coalescing phase, which occurs at the last iteration, when there is an attempt to couple. One-shot coupling does not require the use of any exogenous variables like a drift function or a minorization constant. In this paper, we summarize the one-shot coupling method into the One-Shot Coupling Theorem. We then apply the theorem to two families of Markov chains: the random functional autoregressive process and the autoregressive conditional heteroscedastic (ARCH) process. We provide multiple examples of how the theorem can be used on various models including ones in high dimensions. These examples illustrate how the theorem's conditions can be verified in a straightforward way. The one-shot coupling method appears to generate tight geometric convergence rate bounds.