An exponential inequality for Hilbert-valued U-statistics of i.i.d. data

Abstract

In this paper, we establish an exponential inequality for $U$-statistics of i.i.d. data, varying kernel and taking values in a separable Hilbert space. The bound is expressed as a sum of an exponential term plus an other one involving the tail of a sum of squared norms. We start by the degenerate case. Then we provide applications to $U$-statistics of not necessarily degenerate fixed kernel, incomplete $U$-statistics and weighted $U$-statistics.