Higher Order Concentration of Measure

TL;DR

This paper develops advanced concentration inequalities using higher-order derivatives, applicable to functions of independent variables, measures satisfying Sobolev inequalities, and functions on spheres, with applications to U-statistics and polynomial approximations.

Contribution

It introduces sharpened concentration bounds based on higher-order derivatives, extending classical results to more complex stochastic and geometric settings.

Findings

Derived bounds for deviations of functions of independent variables.

Established concentration inequalities for measures with logarithmic Sobolev properties.

Applied results to U-statistics and polynomial approximations on spheres.

Abstract

We study sharpened forms of the concentration of measure phenomenon typically centered at stochastic expansions of order $d-1$ for any $d \in \mathbb{N}$. The bounds are based on $d$-th order derivatives or difference operators. In particular, we consider deviations of functions of independent random variables and differentiable functions over probability measures satisfying a logarithmic Sobolev inequality, and functions on the unit sphere. Applications include concentration inequalities for $U$-statistics as well as for classes of symmetric functions via polynomial approximations on the sphere (Edgeworth-type expansions).