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April 2003 The $\bm{L}_\mathbf{1}$-norm density estimator process
Evarist Giné, David M. Mason, Andrei Yu. Zaitsev
Ann. Probab. 31(2): 719-768 (April 2003). DOI: 10.1214/aop/1048516534

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 $f$.

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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

Information

Published: April 2003
First available in Project Euclid: 24 March 2003

zbMATH: 1031.62026
MathSciNet: MR1964947
Digital Object Identifier: 10.1214/aop/1048516534

Subjects:
Primary: 60F05 , 60F15 , 60F17 , 62G07

Keywords: $L_{1}$-norm , central limit theorem , Entropy , Kernel density function estimator , poissonization , weak convergeance

Rights: Copyright © 2003 Institute of Mathematical Statistics

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Vol.31 • No. 2 • April 2003
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