  We prove conditional asymptotic normality of a class of quadratic
U-statistics that are dominated by their degenerate second order part and have
kernels that change with the number of observations. These statistics arise in
the construction of estimators in high-dimensional semi- and non-parametric
models, and in the construction of nonparametric confidence sets. This is
illustrated by estimation of the integral of a square of a density or
regression function, and estimation of the mean response with missing data. We
show that estimators are asymptotically normal even in the case that the rate
is slower than the square root of the observations.
