Electronic Journal of Probability

The Law of the Iterated Logarithm for a Triangular Array of Empirical Processes

Miguel Arcones

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Abstract

We study the compact law of the iterated logarithm for a certain type of triangular arrays of empirical processes, appearing in statistics (M-estimators, regression, density estimation, etc). We give necessary and sufficient conditions for the law of the iterated logarithm of these processes of the type of conditions used in Ledoux and Talagrand (1991): convergence in probability, tail conditions and total boundedness of the parameter space with respect to certain pseudometric. As an application, we consider the law of the iterated logarithm for a class of density estimators. We obtain the order of the optimal window for the law of the iterated logarithm of density estimators. We also consider the compact law of the iterated logarithm for kernel density estimators when they have large deviations similar to those of a Poisson process.

Article information

Source
Electron. J. Probab., Volume 2 (1997), paper no. 5, 39 pp.

Dates
Accepted: 18 August 1997
First available in Project Euclid: 26 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1453839981

Digital Object Identifier
doi:10.1214/EJP.v2-19

Mathematical Reviews number (MathSciNet)
MR1475863

Zentralblatt MATH identifier
0888.60010

Subjects
Primary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)
Secondary: 60F15: Strong theorems

Keywords
Empirical process law of the iterated logarithm triangular array density estimation

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Arcones, Miguel. The Law of the Iterated Logarithm for a Triangular Array of Empirical Processes. Electron. J. Probab. 2 (1997), paper no. 5, 39 pp. doi:10.1214/EJP.v2-19. https://projecteuclid.org/euclid.ejp/1453839981


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