Higher order concentration of measure

We study sharpened forms of the concentration of measure phenomenon typically centered at stochastic expansions of order [Formula: see text] for any [Formula: see text]. The bounds are based on [Formula: see text]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 [Formula: see text]-statistics as well as for classes of symmetric functions via polynomial approximations on the sphere (Edgeworth-type expansions).