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 integral(R)f(P)(x)dx < infinity for some p > 2. It is shown, under quite general conditions on the kernel K and on the window sizes, that the centred integrated squared deviation Of f(n),(K) from its mean, \\f(n),(K) - Ef(n),(K) \\(2)(2) - E\\f(n),(K) - Ef(n),(K) \\(2)(2) satisfies a law of the iterated logarithm (LIL). This is then used to obtain an LIL for the, deviation from the true density, \\f(n),(K) - f\\(2)(2) - E\\f(n),(K) - f\\(2)(2). The main tools are the Komlos-Major-Tusnady approximation, a moderate-deviation result for triangular arrays of weighted chi-square variables adapted from work by Pinsky and an exponential inequality due to Gine, Latala and Zinn for degenerate U-statistics applied in combination with decoupling and maximal inequalities.