Best asymptotic normality of the kernel density entropy estimator for smooth densities

In the random sampling setting we estimate the entropy of a probability density distribution by the entropy of a kernel density estimator using the double exponential kernel. Under mild smoothness and moment conditions we show that the entropy of the kernel density estimator equals a sum of independent and identically distributed (i.i.d.) random variables plus a perturbation which is asymptotically negligible compared to the parametric rate n^-1/2. An essential part in the proof is obtained by exhibiting almost sure bounds for the Kullback-Leibler divergence between the kernel density estimator and its expected value. The basic technical tools are Doob´s submartingale inequality and convexity (Jensen´s inequality)