Strong Gaussian approximations in the random truncation mode

In the random left-truncation model, one observes (X-i, Y-i) only if X-i gr eater than or equal to Y-i, i = 1, ..., N. The nonparametric maximum likeli hood estimator aims at reconstructing the distribution function of X from t he observed empirical data. In this paper, strong approximations of the cum ulative hazard process and product-limit process on increasing sets by sequ ences of copies of corresponding Gaussian limiting processes are constructe d. The convergence rates are N-1/6 log N on fixed sets. Futhermore, strong approximations with two-parameter Gaussian processes are obtained with conv ergence rates N-1/8(log N)(3/2) On fixed sets.