Asymptotic normality of the product-limit-estimator

Under the standard right-censorship (RC) model, which assumes that the survival time and censoring time are independent, several sufficient conditions have been established for the product-limit estimator (PLE) being asymptotically normally distributed on the whole real line [see, e.g. Stute, W. (1995), ‘The central limit theorem under random censorship’, Ann. Statist., 23, 422–439]. However, it remains a difficult open problem what the necessary and sufficient conditions that the PLE has an asymptotic normality property on the whole real line is. In this paper, we settle the problem under both the standard RC model which assumes T ⊥ R and the dependent RC model.