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A new characterization of quadratic transportation-information inequalities

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Abstract

It is known that a quadratic transportation-information inequality \(\mathrm{W}_2\mathrm{I}\) interpolates between the Talagrand’s inequality \(\mathrm{W}_2\mathrm{H}\) and the log-Sobolev inequality (LSI for short). The aim of this paper is threefold: (1) To prove the equivalence of \({\mathrm {W}}_2\mathrm{I}\) and the Lyapunov condition, which gives a new characterization inspired by Cattiaux et al. (Probab Theory Relat Fields 148(1–2):285–304, 2010). (2) To prove the stability of \({\mathrm {W}}_2\mathrm{I}\) under bounded perturbations, which gives a transference principle in the sense of Holley–Stroock. (3) To prove \(\mathrm{W}_2\mathrm{H}\) through a restricted \(\mathrm{W}_2\mathrm{I}\), which gives a counterpart of the restricted LSI presented by Gozlan et al. (Ann Probab 39(3):857–880, 2011).

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Acknowledgments

It is my great pleasure to thank Prof. Li-Ming Wu for his warm encouragement. And I deeply appreciate the anonymous reviewer for his/her conscientious reading and many suggestions on the first version. This work is supported by NSFC (No. 11201456, No. 1143000182, No. 11371352), AMSS research grant (No. Y129161ZZ1), and Key Laboratory of Random Complex Structures and Data, Academy of Mathematics and Systems Science, Chinese Academy of Sciences (No. 2008DP173182).

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  1. Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Yuan Liu

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Correspondence to Yuan Liu.

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Liu, Y. A new characterization of quadratic transportation-information inequalities. Probab. Theory Relat. Fields 168, 675–689 (2017). https://doi.org/10.1007/s00440-016-0721-5

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  • DOI: https://doi.org/10.1007/s00440-016-0721-5

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