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. None of the theorems imposes 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.