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 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).
- Concentration of measure phenomenon
- Efron-Stein inequality
- Hoeffding decomposition
- functions on the discrete cube
- logarithmic Sobolev inequalities