Concentration of empirical distribution functions with applications to non-i.i.d. models

TL;DR

This paper investigates how empirical distribution functions concentrate around their expected values in dependent data scenarios, especially for high-dimensional random matrices, using Poincaré and logarithmic Sobolev inequalities.

Contribution

It extends concentration results to dependent data models and applies these findings to spectral empirical distribution functions in high-dimensional matrices.

Findings

Established concentration inequalities for dependent data

Applied results to spectral distribution functions of random matrices

Provided theoretical bounds for empirical measures in complex models

Abstract

The concentration of empirical measures is studied for dependent data, whose joint distribution satisfies Poincar\'{e}-type or logarithmic Sobolev inequalities. The general concentration results are then applied to spectral empirical distribution functions associated with high-dimensional random matrices.