Let Un be a second-degree, nondegenerate, zero mean U-statistic with a bounded kernel, scaled so that Un/[radical sign]n => N(0, [sigma]2). Large deviation approximations are developed for tail probabilities P(Un > x[radical sign]n) using a new explicit tilting procedure.

A large deviation principle for U-statistics is presented. It relies on the large deviation for B-valued r.v. and on the spectral decomposition of the kernel of the U-statistic.