The L1–Norm Density Estimator Process

The notion of an L1–norm density estimator process indexed by a class of kernels is introduced. Then a functional central limit theorem and a Glivenko–Cantelli theorem are established for this process. While assembling the necessary machinery to prove these results, a body of Poissonization techniques and restricted chaining methods is developed, which is useful for studying weak convergence of general processes indexed by a class of functions. This should be of independent interest. None of the theorems impose any condition at all on the underlying Lebesgue density f . Also, somewhat unexpectedly, the distribution of the limiting Gaussian process does not depend on f. AMS Subject Classifications: 60F05, 60F15, 60F17, 62G07