Higher Order Concentration of Measure

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).

some new material and examples added

References in corpus (1)

• Moment inequalities for functions of independent random variables

Cited by in corpus (4)

• Emergent quantum state designs from individual many-body wavefunctions
• Higher order concentration for functions of weakly dependent random variables
• Concentration inequalities for bounded functionals via generalized log-Sobolev inequalities
• Iterated Jackknives and Two-Sided Variance Inequalities