An almost sure invariance principle for Kolmogorov-Smirnov statistics for vector chance variables is established along the lines of Theorems 1.4 and 4.9 of Strassen [Proc. Fifth Berkeley Symp. Math. Statist. Prob. (1967) 2 315-343]. This strengthens certain asymptotic expressions on the probability of moderate deviations for Kolmogorov-Smirnov statistics, obtained earlier by Gnedenko, Karoluk and Skorokhod, and by Kiefer and Wolfowitz, among others.
"An Almost Sure Invariance Principle for Mutivariate Kolmogorov-Smirnov Statistics." Ann. Probab. 1 (3) 488 - 496, June, 1973. https://doi.org/10.1214/aop/1176996943