It is shown that the convergence in law of estimators entails convergence uniformly over all Borel sets whenever the estimators are asymptotically equivariant in a suitable sense and the likelihood ratios of the sample are appropriately smooth. This result generalizes a recent result of Boos in many directions.
"Strong Convergence of Distributions of Estimators." Ann. Statist. 15 (4) 1699 - 1708, December, 1987. https://doi.org/10.1214/aos/1176350619