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Gaussian approximation of local empirical processes indexed by functions. (English) Zbl 0878.60025
The authors introduce a new notion of a local empirical process indexed by functions which extends an earlier version of P. Deheuvels and D. M. Mason [Ann. Probab. 22, No. 3, 1619-1661 (1994; Zbl 0821.60042)], and which includes kernel regression estimators as well as conditional empirical processes. A weak invariance principle is established and also an almost sure Gaussian approximation, based on a suitable coupling inequality. Besides a compact law of the iterated logarithm for such local processes, a number of statistical applications are given which nicely demonstrate the strength of this approach.

60F15 Strong limit theorems
62G05 Nonparametric estimation
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