Open Access
May 2011 Central limit theorems for local empirical processes near boundaries of sets
John H.J. Einmahl, Estáte V. Khmaladze
Bernoulli 17(2): 545-561 (May 2011). DOI: 10.3150/10-BEJ283


We define the local empirical process, based on $n$ i.i.d. random vectors in dimension $d$, in the neighborhood of the boundary of a fixed set. Under natural conditions on the shrinking neighborhood, we show that, for these local empirical processes, indexed by classes of sets that vary with $n$ and satisfy certain conditions, an appropriately defined uniform central limit theorem holds. The concept of differentiation of sets in measure is very convenient for developing the results. Some examples and statistical applications are also presented.


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John H.J. Einmahl. Estáte V. Khmaladze. "Central limit theorems for local empirical processes near boundaries of sets." Bernoulli 17 (2) 545 - 561, May 2011.


Published: May 2011
First available in Project Euclid: 5 April 2011

zbMATH: 1248.60030
MathSciNet: MR2787604
Digital Object Identifier: 10.3150/10-BEJ283

Keywords: convex body , differentiation of sets , Gaussian behavior , local empirical process , set boundary , weak convergence

Rights: Copyright © 2011 Bernoulli Society for Mathematical Statistics and Probability

Vol.17 • No. 2 • May 2011
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