Extension to higher dimensions of the Jaeschke-Eicker result on the standardized empirical process

The asymptotic distribution of the sup-norm of the heavily weighted empirical process is established in the multidimensional case. This theorem extends in particular the famous result in Jaeschke (1975,1979) to higher dimensions. There is a striking difference between the behaviour for higher dimensions and that for dimension one, especially the limiting distribution is now a simple transformation of a standard exponential random variable. Key words and phrases: Asymptotic distribution, empirical process, heavy weights.