Limit theorems and probability inequalities for $U$-statistics

The study of the asymptotic behavior of U-statistics has concentrated a lot of attention since the seminal work by Hoeffding (1963), due to its applications in particular to statistics and random geometry. In this talk, we will review some results on U-statistics of i.i.d. data (Hoeffding's decomposition, law of large numbers, law of the iterated logarithms, central limit theorem) and give an exponential inequality for U-statistics of i.i.d. data and furnish applications to the rates in the strong law of large numbers and functional central limit theorems in Hölder spaces. Then we will deal with a generalized Hoeffding's decomposition for U-statistics whose data is a function of i.i.d. random variables.