GAUSSIAN APPROXIMATION OF LOCAL EMPIRICAL PROCESSES INDEXED BY FUNCTIONS

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.