A new method for proving weak convergence results applied to nonparametric estimators in survival analysis

Using the limit theorem for stochastic integral obtained by Jakubowski et al. (Probab. Theory Related Fields 81 (1989) 111-137), we introduce in this paper a new method for proving weak convergence results of empirical processes by a martingale method which allows discontinuities for the underlying distribution. This is applied to Nelson-Aalen and Kaplan-Meier processes. We also prove that the same conclusion can be drawn for Hjort's nonparametric Bayes estimators of the cumulative distribution function and cumulative hazard rate.