Strong Approximations to the Local Empirical Process

Abstract

In a previous paper we introduced a notion of a local empirical process indexed by functions which has useful applications to density and regression function estimation among other areas. We have shown that if the function class is uniformly bounded, one can obtain a strong approximation to this process by a suitable Gaussian process. We now prove such a result when the underlying function class is unbounded, but has an envelope function with a finite p-th moment for some p > 2. Among other applications, our new strong invariance principle for the unbounded case can be used to prove laws of the iterated logarithm for the kernel regression function estimator under mild conditions.KeywordsEmpirical ProcessReproduce Kernel Hilbert SpaceIterate LogarithmStrong ApproximationKernel Density EstimatorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.