  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).
