Some Selected Topics for the Bootstrap of the Empirical and Quantile processes

In the present work, we consider the asymptotic distributions of $L_{p}$ functionals of bootstrapped weighted uniform quantile and empirical processes. The asymptotic laws obtained are represented in terms of Gaussian integrals. We investigate the strong approximations for the bootstrapped Vervaat process and the weighted bootstrap for Bahadur-Kiefer process. We obtain new results on the precise asymptotics in the law of the logarithm related to complete convergence and a.s. convergence, under some mild conditions, for the weighted bootstrap of empirical and the quantile processes. In addition we consider the strong approximation of the hybrids of empirical and partial sums processes when the sample size is random.