U. Einmahl et Dm. Mason, GAUSSIAN APPROXIMATION OF LOCAL EMPIRICAL PROCESSES INDEXED BY FUNCTIONS, Probability theory and related fields, 107(3), 1997, pp. 283-311
An extended notion of a local empirical process indexed by functions i
s introduced, which includes kernel density and regression function es
timators and the conditional empirical process as special cases. Under
suitable regularity conditions a central limit theorem and a strong a
pproximation by a sequence of Gaussian processes are established for s
uch processes. A compact law of the iterated logarithm (LIL) is then i
nferred from the corresponding LIL for the approximating sequence of G
aussian processes. A number of statistical applications of our results
are indicated.