Noncommutative martingale deviation and Poincaré type inequalities with applications

Marius Junge, Qiang Zeng*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Scopus citations

Abstract

We prove a deviation inequality for noncommutative martingales by extending Oliveira’s argument for random matrices. By integration we obtain a Burkholder type inequality with satisfactory constant. Using continuous time, we establish noncommutative Poincaré type inequalities for “nice” semigroups with a positive curvature condition. These results allow us to prove a general deviation inequality and a noncommutative transportation inequality due to Bobkov and Götze in the commutative case. To demonstrate our setting is general enough, we give various examples, including certain group von Neumann algebras, random matrices and classical diffusion processes, among others.

Original languageEnglish (US)
Pages (from-to)449-507
Number of pages59
JournalProbability Theory and Related Fields
Volume161
Issue number3-4
DOIs
StatePublished - Apr 1 2015

Keywords

  • (Noncommutative) Burkholder inequality
  • (Noncommutative) Burkholder–Davis–Gundy inequality
  • (Noncommutative) Poincaré inequality
  • (Noncommutative) diffusion processes
  • (Noncommutative) martingale deviation inequality
  • (Noncommutative) transportation inequality
  • Concentration inequality
  • Group von Neumann algebras
  • Noncommutative L spaces
  • Γ-Criterion

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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