Random weighting approximation for empirical processes and their applications*

In this note, the random weighting approximations for empirical process indexed by abounded class of functions are obtained, and the strong uniform convergence rates of ran-dom weighting empirical processes are given. When unknown multivariate underlingdistributions are not confined by any additional condition, we obtain the random weightingapproximations for multivariate Von Mises statistics and their projection pursuits (PP); thisresult is true for the bootstrap approximation of empirical processes, too.