2007/07/01 by Michael Aizenman, François Germinet, Francois Germinet +2
Engineering · Mathematics · Physics and Astronomy · #Bernoulli distribution #Bernoulli process #Bernoulli scheme #Bernoulli's principle #Probabilistic logic #Probability theory #Random Matrices and Applications #Random element #Random function #Random variable #Spectral Theory in Mathematical Physics #Stochastic process #Wireless Communication Security Techniques #math-ph #math.MP #math.PR
paper · pdf · doi:10.1007/s00440-007-0125-7
published as Probab. Theory Relat. Fields (2009) 143: 219-238
arxiv created 2007/07/01 · openalex publication_date 2008/01/29 · arxiv updated 2010/10/26 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06
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 Schroedinger 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.