Multivariate integral kernel transformations of the multivariate empirical process are considered. The asymptotic behaviour of these transforms are investigated when a null-hypothesis completely specifies the underlying distribution and also when parameters are also estimated from the sample. In both cases conditions for the kernel are found ensuring that strong approximation, weak convergence, rates of convergence and uniform consistency results hold, the latter often being in the form of functional and common log log laws. It is hoped that certain composite hypothesis-testing problems might be handled by the obtained results. A number of problems are posed in this regard.
