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Second and higher order concentration of measure

Banff International Research Station for Mathematical Innovation and Discovery

Higher order concentration results are proved for differentiable functions on Euclidean spaces with LSI type measures provided that they are Lipschitz bounded of order $d\ge 2$ and orthogonal to polynomials of order $d-1$. This is recent joint work with S. Bobkov and H. Sambale. It extends to concentration of measure for functions on discrete spaces subject to higher order $L^2$ type differences uniformly bounded and an appropriate Hoeffding expansion structure. The results yield uniform exponential bounds for $|f|^{2/d}$ extending previous 2nd order results for functions on these spaces. Some applications of these bounds are given. In particular applications of 2nd order concentrations for polynomials on the sphere and spherical averages of 2nd order uncorrelated isotropic vector are discussed, illustrating previous joint results with S.Bobkov and G. Chistyakov.

Mathematics; Probability theory and stochastic processes; Statistics; Probability / statistical mechanics

video/mp4

Author affiliation: University of Bielefeld

2019-03-11

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/68591

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/