In this paper, stochastic processes developed by Aalen   are adapted to the Nelson-Aalen and Kaplan-Meier  estimators in a context of competing risks. We focus only on the probability distributions of complete downtime individuals whose causes are known and which bring us to consider a partition of individuals into sub-groups for each cause. We then study the asymptotic properties of nonparametric estimators obtained.
 Aalen, O.O. (1978) Nonparametric Estimation of Partial Transition Probabilities in Multiple Decrement Models. The Annals of Statistics, 6, 534-545. http://dx.doi.org/10.1214/aos/1176344198
 Aalen, O.O. (1978) Nonparametric Inference for a Family of Counting Processes. The Annals of Statistics, 6, 701-726.
 Kaplan, E.L. and Meier, P. (1958) Nonparametric Estimation from Incomplete Observations. Journal of the American Statistical Association, 53, 457-481. http://dx.doi.org/10.1080/01621459.1958.10501452