Abstract Consider a long term study, where a series of possibly censored failure times is observed. Suppose the failure times have a common marginal distribution function F, but they exhibit a mode of dependence characterized by positive or negative association. Under suitable regularity conditions, it is shown that the Kaplan–Meier estimator F n of Fis uniformly strongly consistent; rates for the convergence are also provided. Similar results are established for the empirical cumulative hazard rate function involved. Furthermore, a stochastic process generated by F n is shown to be weakly convergent to an appropriate Gaussian process. Finally, an estimator of the limiting variance of the Kaplan–Meier estimator is proposed and it is shown to be weakly convergent.