The Law of the Iterated Logarithm for the Integrated Squared Deviation of a Kernel Density Estimator

Let f n,K denote a kernel estimator of a density f in R such that R f p (x)dx<∞ for some p>2. It is shown, under quite general conditions on the kernel K and on the window sizes, that the centered integrated squared deviation of f n,K from its mean, f n,K −Ef n,K 2 2 −Ef n,K −Ef n,K 2 2 satisfies a law of the iterated logarithm. This is then used to obtain an LIL for the deviation from the true density, f n,K −f 2 2 −Ef n,K −f 2 2. The main tools are the Komlós-Major-Tusnády approximation, a moderate deviation result for triangular arrays of weighted chi-square variables adapted from Pinsky (1966) and an exponential inequality of Giné, Lataa la and Zinn (2000) for degenerate U-statistics applied in combination with decoupling and maximal inequalities. Runninghead: The LIL for L 2 functionals of kernel density estimators 1. Introduction. As far as we know there are no laws of the iterated logarithm for the integrated p-th absolute deviation of a kernel density estimator from its mean, although central limit theorems do exist This anomaly seems to be due to the fact that there are serious difficulties both to find the proper way in blocking and in deriving sufficiently precise moderate deviation results. In this paper we show how these difficulties can be handled in the case p = 2. In the process we shall spotlight a number of techniques that should be of independent interest. Unfortunately our methods do not extend to other values of p. In order to make our aim clear let us now fix some notation and introduce our basic assumptions. Throughout this paper we shall assume that f is a probability density on the real line R such that