Abstract
In this paper, one investigates the transportation-information T c I inequalities: α(T c (ν, μ)) ≤ I (ν|μ) for all probability measures ν on a metric space \({(\mathcal{X}, d)}\), where μ is a given probability measure, T c (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 T c I 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 T c I and comparisons of T c I with Poincaré and log-Sobolev inequalities are investigated. Several easy-to-check sufficient conditions are provided for special important cases of T c I and several examples are worked out.
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Guillin, A., Léonard, C., Wu, L. et al. Transportation-information inequalities for Markov processes. Probab. Theory Relat. Fields 144, 669–695 (2009). https://doi.org/10.1007/s00440-008-0159-5
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DOI: https://doi.org/10.1007/s00440-008-0159-5