Transportation-information inequalities for Markov processes

By Arnaud Guillin, Christian Léonard, Liming Wu, Nian Yao. 2007

111 citations recorded by the source corpus.

In this paper, one investigates the transportation-information TcI inequalities: α(Tc(ν, μ)) ≤ I (ν|μ) for all probability measures ν on a metric space $${(\mathcal{X}, d)}$$, where μ is a given probability measure, Tc(ν, μ) is the transportation cost from ν to μ with respect to the cost function c(x, y) on $${\mathcal{X}^2}$$, I(ν|μ) is the Fisher–Donsker–Varadhan information of ν with respect to μ and α : [0, ∞) → [0, ∞] is a left continuous increasing function. Using large deviation techniques, it is shown that TcI is equivalent to some concentration inequality for the occupation measure of a μ-reversible ergodic Markov process related to I(·|μ). The tensorization property of TcI and comparisons of TcI with Poincaré and log-Sobolev inequalities are investigated. Several easy-to-check sufficient conditions are provided for special important cases of TcI and several examples are worked out.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8466b4d4240b57c8f0a12f0abeda1286d5a56dbd

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Citations in the corpus

A new Poisson-type deviation inequality for Markov jump processes with positive Wasserstein curvature (2009, background): “See also the recent work of Guillin et al. (2008) for an approach to this problem through transportation-information inequalities.”
Convergence rates for rank-based models with applications to portfolio theory (2011, methodology): “[19], we obtain the estimate of Theorem 1 for any bounded measurable function of the particle configuration given by the components of the process Ỹ .”
Semi Log-Concave Markov Diffusions (2013, background): “4 or [29] for examples and details on properties of WI inequality).”
Curvature and transport inequalities for Markov chains in discrete spaces (2015, background): “When p = 1, the transport-information inequality is equivalent to Gaussian concentration for the occupation measure of the Markov chain (see [21]), and is therefore useful to get a priori bounds on the statistical error for Markov Chain Monte Carlo estimation of averages.”
Transport Inequalities. A Survey (2010, methodology): “6 (W2I in relation with LS and P, [57]).”
A characterization of dimension free concentration in terms of transportation inequalities (2008, result): “We think that this new result confirms the relevance of the Large Deviations point of view for functional inequalities initiated by Léonard and the author in [22] and pursued in [24] by Guillin et al.”