On Bernoulli decompositions for random variables, concentration bounds, and spectral localization

Michael Aizenman, François Germinet, Abel Klein, Simone Warzel

Research output: Contribution to journalArticle

15 Scopus citations

Abstract

As was noted already by A. N. Kolmogorov, any random variable has a Bernoulli component. This observation provides a tool for the extension of results which are known for Bernoulli random variables to arbitrary distributions. Two applications are provided here: (i) an anti-concentration bound for a class of functions of independent random variables, where probabilistic bounds are extracted from combinatorial results, and (ii) a proof, based on the Bernoulli case, of spectral localization for random Schrödinger operators with arbitrary probability distributions for the single site coupling constants. For a general random variable, the Bernoulli component may be defined so that its conditional variance is uniformly positive. The natural maximization problem is an optimal transport question which is also addressed here.

Original languageEnglish (US)
Pages (from-to)219-238
Number of pages20
JournalProbability Theory and Related Fields
Volume143
Issue number1-2
DOIs
StatePublished - Jan 2009

All Science Journal Classification (ASJC) codes

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

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