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An Introduction to Coupling

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Modeling, Dynamics, Optimization and Bioeconomics II (DGS 2014)

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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 \(\mathscr {P}\) has all entries positive.

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Notes

  1. 1.

    \(\varLambda =1\) if and only if J is constant on cylinders of length 1.

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Author information

Authors and Affiliations

  1. Instituto de Matemática-UFRGS, Avenida Bento Gonçalves 9500, Porto Alegre-RS, Brazil

    Artur O. Lopes

Authors
  1. Artur O. Lopes
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Correspondence to Artur O. Lopes .

Editor information

Editors and Affiliations

  1. Department of Mathematics and LIAAD-INESC TEC, University of Porto, Porto, Portugal

    Alberto A. Pinto

  2. Department of Agricultural and Resource Economics, University of California, Berkeley, California, USA

    David Zilberman

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Lopes, A.O. (2017). An Introduction to Coupling . In: Pinto, A., Zilberman, D. (eds) Modeling, Dynamics, Optimization and Bioeconomics II. DGS 2014. Springer Proceedings in Mathematics & Statistics, vol 195. Springer, Cham. https://doi.org/10.1007/978-3-319-55236-1_15

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