The Annals of Probability

The $\bm{L}_\mathbf{1}$-norm density estimator process

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 $f$.

Article information

Source
Ann. Probab. Volume 31, Number 2 (2003), 719-768.

Dates
First available in Project Euclid: 24 March 2003

https://projecteuclid.org/euclid.aop/1048516534

Digital Object Identifier
doi:10.1214/aop/1048516534

Mathematical Reviews number (MathSciNet)
MR1964947

Zentralblatt MATH identifier
1031.62026

Citation

Giné, Evarist; Mason, David M.; Zaitsev, Andrei Yu. The $\bm{L}_\mathbf{1}$-norm density estimator process. Ann. Probab. 31 (2003), no. 2, 719--768. doi:10.1214/aop/1048516534. https://projecteuclid.org/euclid.aop/1048516534

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• STORRS, CONNECTICUT 06269-3009 E-MAIL: gine@uconnvm.uconn.edu D. M. MASON DEPARTMENT OF FOOD AND RESOURCE ECONOMICS 206 TOWNSEND HALL UNIVERSITY OF DELAWARE
• NEWARK, DELAWARE 19717 E-MAIL: davidm@udel.edu A. YU. ZAITSEV LABORATORY OF STATISTICAL METHODS ST. PETERSBURG BRANCH OF THE STEKLOV MATHEMATICAL INSTITUTE 27 FONTANKA ST. PETERSBURG 191011 RUSSIA E-MAIL: zaitsev@pdmi.ras.ru