Moderate deviations and law of the iterated logarithm in for kernel density estimators

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