The cumulative quantile regression function with censored and truncated covariate

In this paper, we consider the empirical estimator of the cumulative quantile regression (CQR) functionwhen the covariate is subjected to random truncation and censorship. Strong Gaussian approximations for the associated CQR process are established under appropriate assumptions. A functional law of the iterated logarithm for the CQR process is also derived. These results provide a foundation for the asymptotic theory of functional statistics based on these processes.