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 ∈ 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).

Iterated Jackknives and Two-Sided Variance Inequalities

“…1 also easily imply the following corollary obtained in[1] (as part of their Theorem 1.8) which moreover can be complemented with the trivial lower bound EK d /k! ≤ Var S.…”

“…(iii) As in [2] or [1], one could also rewrite (2.8) using only the positive or negative parts of the involved quantities.…”

“…The aim of the present note is to provide a synthetic proof of these, removing the iid and symmetry assumptions in (1.5) and its generalizations, leading to generic inequalities. This could be useful, as these dormant inequalities seem to have found, in recent times, some new life, e.g., see [8], [1], [9].…”

Explicit rates of convergence in the multivariate CLT for nonlinear statistics

“…Following the notations introduced in [4], we recall the definition of two certain difference operators which will be used in our work.…”

“…Let us now recall some useful properties of the operators D and d, see e.g. [4,15]. For the sake of completeness we will give a brief proof of those properties.…”