AN EXPONENTIAL INEQUALITY FOR U-STATISTICS OF IID DATA

We establish an exponential inequality for degenerated U-statistics of order r of independent and identically distributed (i.i.d.) data. This inequality gives a control of the tail of the maxima absolute values of the U-statistic by the sum of the two terms: an exponential term and one involving the tail of h(X1, ... , Xr). We also give a version for not necessarily degenerated U-statistics having a symmetric kernel and furnish an application to the convergence rates in the Marcinkiewicz law of large numbers. Application to the invariance principle in Ho center dot lder spaces is also considered.