In this paper we derive a law of the logarithm for the maximal deviation between a kernel density estimator and the true underlying density function. Extensions to higher derivatives are included. The results are applied to get optimal window-widths with respect to almost sure uniform convergence.
"A Law of the Logarithm for Kernel Density Estimators." Ann. Probab. 10 (2) 414 - 422, May, 1982. https://doi.org/10.1214/aop/1176993866