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.
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 language | English |
|---|---|
| Pages (from-to) | 1952-1970 |
| Number of pages | 19 |
| Journal | Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques |
| Volume | 53 |
| Issue number | 4 |
| Early online date | 27 Nov 2017 |
| DOIs | |
| Publication status | Published - 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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Profiles
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Professor Oliver T Johnson
- School of Mathematics - Professor of Information Theory
- Statistical Science
- Probability, Analysis and Dynamics
- Probability
Person: Academic , Academic , Member