On Bernoulli Decompositions for Random Variables, Concentration Bounds, and Spectral Localization

TL;DR

This paper explores Bernoulli decompositions of arbitrary random variables to extend concentration bounds and spectral localization results, providing new tools for probabilistic and spectral analysis.

Contribution

It introduces a method to decompose any random variable into Bernoulli components, enabling the extension of known Bernoulli results to general distributions.

Findings

Derived anti-concentration bounds for functions of independent variables.

Proved spectral localization for random Schrödinger operators with arbitrary distributions.

Addressed an optimal transport problem related to Bernoulli decompositions.

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 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.