Abstract
The notion of an $L_{1}$-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 $f$.
Citation
Evarist Giné. David M. Mason. Andrei Yu. Zaitsev. "The $\bm{L}_\mathbf{1}$-norm density estimator process." Ann. Probab. 31 (2) 719 - 768, April 2003. https://doi.org/10.1214/aop/1048516534
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