Moderate deviations and law of the iterated logarithm in L1(Rd)L1(Rd) for kernel density estimators

Let fn(x)fn(x) be the non-parametric kernel density estimator of a density function f(x)f(x) based on a kernel function KK. In this paper, we first prove two moderate deviation theorems in L1(Rd)L1(Rd) for {fn(x),n≥1}{fn(x),n≥1}. Then, as an application of the moderate deviations, we obtain a law of the iterated logarithm for {‖fn−Efn‖1,n≥1}{‖fn−Efn‖1,n≥1}.