That the Kolmogorov-Smirnov statistics obey iterated logarithm laws is well known. For the purpose of developing nonparametric tests with power one it has become of interest to find accurate upper bounds for the probability that a sequence of Kolmogorov-Smirnov statistics ever exceeds a given boundary sequence. This paper is concerned with finding such probability bounds for a wide class of boundary sequences.
Richard M. Stanley. "Boundary Crossing Probabilities for the Kolmogorov-Smirnov Statistics." Ann. Math. Statist. 43 (2) 664 - 668, April, 1972. https://doi.org/10.1214/aoms/1177692650