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Higher order concentration of measure

Communications in Contemporary Mathematics. 2019. Vol. 21. No. 3. P. 1-31.
Bobkov S., Goetze F., Sambale H.

We study sharpened forms of the concentration of measure phenomenon typically centered at stochastic expansions of order d-1 for any d \in N. The bounds are based on dth 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).