For series of independent random processes in the space $D\lbrack 0, 1 \rbrack$ endowed with the Skorohod topology, convergence in distribution is shown to imply almost sure convergence. Under mild conditions, such as e.g. when the limiting process has no jumps of fixed size and location, the latter convergence is uniform. As an application, we discuss a representation by Ferguson and Klass of processes with independent increments.
"Series of Random Processes without Discontinuities of the Second Kind." Ann. Probab. 2 (4) 729 - 737, August, 1974. https://doi.org/10.1214/aop/1176996615