An introduction to coupling

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 the exponential convergence to equilibrium on the state space for finite state Markov chains when the transition matrix P has all entries positive.