PUB - Publikationen an der Universität Bielefeld

In this paper, we prove multilevel concentration inequalities for bounded functionals f = f ( X-1,..., X-n) of random variables X-1,..., 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) = supg is an element of F vertical bar g(X)vertical bar and suprema of homogeneous chaos in bounded random variables in theBanach space case f (X) = sup(t)parallel to Sigma(i1 not equal)...not equal(id) t(i1)... (i)d X-i1 ... X-id parallel to(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 forU-statistics with bounded kernels h and for the number of triangles in an exponential random graph model.