Let $T$ be the survival time, $R$ be the censoring time and $S(t)=P(T>t)$. If $T$ and $R$ are independent ($T\perp R$), several sufficient conditions have been established for the product-limit estimator (PLE) being asymptotically normally distributed on the whole real line (see, {\it e.g.}, Stute (1995)). However, the necessary and sufficient condition for the PLE to have an asymptotic normality property on the whole real line remains a difficult open problem. In this paper, we settle the problem under both the standard RC model which assumes $T\perp R$ and the dependent RC model.