The rate of strong uniform consistency for the multivariate product-limit estimator

The general asymptotic order of magnitude is determined for the maximal deviation of the multivariate product-limit estimate from the estimated survival function on R^k. This order depends on the joint behavior of the censoring and censored distributions in a well-defined way. Corresponding to specific joint behaviors, several lim sup results are deduced generalizing everything that is known in the univariate case. The results are also extended for the variable censoring model.