  A common statistical task lies in showing asymptotic normality of certain
statistics. In many of these situations, classical textbook results on weak
convergence theory suffice for the problem at hand. However, there are quite
some scenarios where stronger results are needed in order to establish an
asymptotic normal approximation uniformly over a family of probability
measures. In this note we collect some results in this direction. We restrict
ourselves to weak convergence in $\mathbb R^d$ with continuous limit measures.
