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On Bernoulli decompositions for random variables, concentration bounds, and spectral localization
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  • Published: 30 January 2008

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

  • Michael Aizenman1,
  • François Germinet2,
  • Abel Klein3 &
  • …
  • Simone Warzel4 

Probability Theory and Related Fields volume 143, pages 219–238 (2009)Cite this article

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  • 19 Citations

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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.

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Authors and Affiliations

  1. Departments of Mathematics and Physics, Princeton University, Princeton, NJ, 08544, USA

    Michael Aizenman

  2. Département de Mathématiques, Université de Cergy-Pontoise, Laboratoire AGM, UMR CNRS 8088, Institut universitaire de France, 2 avenue Adolphe Chauvin, 95302, Cergy-Pontoise Cedex, France

    François Germinet

  3. Department of Mathematics, University of California at Irvine, Irvine, CA, 92697-3875, USA

    Abel Klein

  4. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Simone Warzel

Authors
  1. Michael Aizenman
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  2. François Germinet
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  3. Abel Klein
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  4. Simone Warzel
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Correspondence to Michael Aizenman.

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Aizenman, M., Germinet, F., Klein, A. et al. On Bernoulli decompositions for random variables, concentration bounds, and spectral localization. Probab. Theory Relat. Fields 143, 219–238 (2009). https://doi.org/10.1007/s00440-007-0125-7

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  • Received: 29 June 2007

  • Revised: 03 December 2007

  • Published: 30 January 2008

  • Issue Date: January 2009

  • DOI: https://doi.org/10.1007/s00440-007-0125-7

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Mathematics Subject Classification (2000)

  • 60E15
  • 82B44
  • 06A07
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