An exponential inequality for $U$-statistics of i.i.d. 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 $h(X_1,\dots,X_r)$. 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 Hölder spaces is also considered.