If a sequence of distributions converges weakly to a distribution and if the associated densities are unimodal then the sequence of density functions converges to the density of the limiting distribution. This stronger form of convergence is useful in the investigation of limiting joint distributions.
"A Note on the Strong Convergence of Distributions." Ann. Statist. 2 (3) 597 - 598, May, 1974. https://doi.org/10.1214/aos/1176342722