A discrete log-Sobolev inequality under a Bakry-Émery type condition

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Abstract

We consider probability mass functions V supported on the positive integers using
arguments introduced by Caputo, Dai Pra and Posta, based on a Bakry–Émery condition for a Markov birth and death operator with invariant measure V . Under this condition, we prove a new modified logarithmic Sobolev inequality, generalizing and strengthening results of Wu, Bobkov and Ledoux, and Caputo, Dai Pra and Posta. We show how this inequality implies results including concentration of measure and hypercontractivity, and discuss how it may extend to higher dimensions.
Original languageEnglish
Pages (from-to)1952-1970
Number of pages19
JournalAnnales de l'Institut Henri Poincaré (B) Probabilités et Statistiques
Volume53
Issue number4
Early online date27 Nov 2017
DOIs
Publication statusPublished - Nov 2017

Keywords

  • Bakry–Émery condition
  • Birth and death chain
  • Concentration of measure
  • Discrete probability measure
  • Log-concavity
  • Log-Sobolev inequality

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