We prove functional laws of the iterated logarithm for empirical processes based upon censored data in the neighborhood of a fixed point. We apply these results to obtain strong laws for estimators of local functionals of the lifetime distribution. In particular, we describe the pointwise strong limiting behavior of the kernel density estimator based upon the Kaplan-Meier product-limit estimator.
Einmahl, J. H. J., & Deheuvels, P. (1996). On the strong limiting behavior of local functionals of empirical processes based upon censored data. The Annals of Probability, 24(1), 504-525. https://doi.org/10.1214/aop/1042644729