Summary.
An extended notion of a local empirical process indexed by functions is introduced, which includes kernel density and regression function estimators and the conditional empirical process as special cases. Under suitable regularity conditions a central limit theorem and a strong approximation by a sequence of Gaussian processes are established for such processes. A compact law of the iterated logarithm (LIL) is then inferred from the corresponding LIL for the approximating sequence of Gaussian processes. A number of statistical applications of our results are indicated.
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Received: 11 January 1995/In revised form: 12 July 1996
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Einmahl, U., Mason, D. Gaussian approximation of local empirical processes indexed by functions. Probab Theory Relat Fields 107, 283–311 (1997). https://doi.org/10.1007/s004400050086
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DOI: https://doi.org/10.1007/s004400050086