Abstract
In this paper, we prove multilevel concentration inequalities for bounded functionals f = f(X_1, ldots , X_n) of random variables X_1, ldots , X_n that are either independent or satisfy certain logarithmic Sobolev inequalities. The constants in the tail estimates depend on the operator norms of k-tensors of higher order differences of f. We provide applications for both dependent and independent random variables. This includes deviation inequalities for empirical processes f(X) = sup _{g in {mathcal {F}}} {|g(X)|} and suprema of homogeneous chaos in bounded random variables in the Banach space case f(X) = sup _{t} {Vert sum _{i_1 ne ldots ne i_d} t_{i_1 ldots i_d} X_{i_1} cdots X_{i_d}Vert }_{{mathcal {B}}}. The latter application is comparable to earlier results of Boucheron, Bousquet, Lugosi, and Massart and provides the upper tail bounds of Talagrand. In the case of Rademacher random variables, we give an interpretation of the results in terms of quantities familiar in Boolean analysis. Further applications are concentration inequalities for U-statistics with bounded kernels h and for the number of triangles in an exponential random graph model.
Highlights
During the last forty years, the concentration of measure phenomenon has become an established part of probability theory with applications in numerous fields, as is witnessed by the monographs [18,38,42,45,54]
One way to prove concentration of measure is by using functional inequalities, the entropy method
By means of a multilevel concentration inequality, we can show that while for t large, subexponential tail decay holds, for small t we even get subgaussian decay
Summary
During the last forty years, the concentration of measure phenomenon has become an established part of probability theory with applications in numerous fields, as is witnessed by the monographs [18,38,42,45,54]. This is already obvious if we consider the product of two independent standard normal random variables, which leads to subexponential tails We refer to this topic as higher order concentration. Multilevel concentration inequalities have been proven in [1,5,56] for many classes of functions These included U -statistics in independent random variables, functions of random vectors satisfying Sobolev-type inequalities and polynomials in sub-Gaussian random variables, respectively. As multilevel or higher order (d-th order) concentration inequalities This means that the tails might have different decay properties in some regimes of [0, ∞). We provide improvements of earlier higher order concentration results like [10, Theorem 1.1] or [28, Theorem 1.5], replacing the Hilbert–Schmidt norms appearing therein by operator norms This leads to sharper bounds and a wider range of applicability.
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