  This paper investigates weighted approximations for studentized
$U$-statistics type processes, both with symmetric and antisymmetric kernels,
only under the assumption that the distribution of the projection variate is in
the domain of attraction of the normal law. The classical second moment
condition $E|h(X_1,X_2)|^2 < \infty$ is also relaxed in both cases. The results
can be used for testing the null assumption of having a random sample versus
the alternative that there is a change in distribution in the sequence.
