Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An introduction to Coupling

Artur O. Lopes

TL;DR

The article surveys coupling techniques in ergodic theory and probability, with a focus on contracting the dual Ruelle operator in the $1$-Wasserstein distance and on exponential convergence to equilibrium for finite-state Markov chains with strictly positive transitions. It builds a framework using maximal couplings, random and stopping times, and the $\\bar{d}$ distance to derive quantitative convergence bounds. A central result is that the dual of the Ruelle operator satisfies a contraction in the $1$-Wasserstein metric after a suitable time, under Lipschitz and bounded distortion conditions, yielding uniqueness and exponential convergence of Gibbs-type states. The work connects dynamical systems methods with probabilistic coupling techniques to provide robust tools for decay of correlations and equilibrium analysis across Bernoulli shifts and Gibbs measures.

Abstract

In this review paper, we describe the use of couplings in several different mathematical problems. We consider the total variation norm, maximal coupling, and the $\bar{d}$-distance. We present a detailed proof of a result recently proved: the dual of the Ruelle operator is a contraction with respect to $1$-Wasserstein distance. We also show exponential convergence to equilibrium in the state space for finite-state Markov chains when the transition matrix $\mathcal{P}$ has all entries positive.} In this new version, we describe in more detail the line of reasoning followed in the work previously published as a chapter in ``Modeling, Dynamics, Optimization and Bioeconomics II'', Springer Verlag (2017).

An introduction to Coupling

TL;DR

The article surveys coupling techniques in ergodic theory and probability, with a focus on contracting the dual Ruelle operator in the -Wasserstein distance and on exponential convergence to equilibrium for finite-state Markov chains with strictly positive transitions. It builds a framework using maximal couplings, random and stopping times, and the distance to derive quantitative convergence bounds. A central result is that the dual of the Ruelle operator satisfies a contraction in the -Wasserstein metric after a suitable time, under Lipschitz and bounded distortion conditions, yielding uniqueness and exponential convergence of Gibbs-type states. The work connects dynamical systems methods with probabilistic coupling techniques to provide robust tools for decay of correlations and equilibrium analysis across Bernoulli shifts and Gibbs measures.

Abstract

In this review paper, we describe the use of couplings in several different mathematical problems. We consider the total variation norm, maximal coupling, and the -distance. We present a detailed proof of a result recently proved: the dual of the Ruelle operator is a contraction with respect to -Wasserstein distance. We also show exponential convergence to equilibrium in the state space for finite-state Markov chains when the transition matrix has all entries positive.} In this new version, we describe in more detail the line of reasoning followed in the work previously published as a chapter in ``Modeling, Dynamics, Optimization and Bioeconomics II'', Springer Verlag (2017).

Paper Structure

This paper contains 9 sections, 16 theorems, 248 equations.

Table of Contents

  1. Introduction
  2. Some definitions and the dual of the Ruelle Operator
  3. The total variation norm and the maximal coupling
  4. Estimates using the random time $T$
  5. Exponential convergence to equilibrium for finite state Markov chains
  6. Estimates using the stopping time $T_1$
  7. Speed estimation on total variation
  8. The $\bar{d}$ distance
  9. Contraction for the dual of the Ruelle Operator

Key Result

Theorem 1

Denote by $Y_n^\nu$, $n \in \mathbb{N}$, the Markov probability associated with the stochastic matrix $\mathcal{P}$, with initial vector of probability $\nu$. Then, for any $n$ we have and this describes the speed of convergence to the equilibrium $\lambda$, when time goes to infinity, for the Markov Process $Y_n^\nu$ .

Theorems & Definitions (32)

  • Theorem 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Proposition 1
  • Theorem 2
  • Proposition 2
  • ...and 22 more