  The convergence of U-statistics has been intensively studied for estimators
based on families of i.i.d. random variables and variants of them. In most
cases, the independence assumption is crucial [Lee90, de99]. When dealing with
Feynman-Kac and other interacting particle systems of Monte Carlo type, one
faces a new type of problem. Namely, in a sample of N particles obtained
through the corresponding algorithms, the distributions of the particles are
correlated -although any finite number of them is asymptotically independent
with respect to the total number N of particles. In the present article,
exploiting the fine asymptotics of particle systems, we prove convergence
theorems for U-statistics in this framework.
