I consider the calculation of the probability $P_n$ that the graph of a sample distribution function lie wholly to one side of a given arbitrary contour. A generating function approach is described in Section 2, and $P_n$ calculated exactly for some simple types of contour. Upper and lower bounds of the correct asymptotic form (relations (14), (15)) are obtained for $P_n$ in the case of a straight line contour.
"Some Exact Results for One-Sided Distribution Tests of the Kolmogorov-Smirnov Type." Ann. Math. Statist. 32 (2) 499 - 505, June, 1961. https://doi.org/10.1214/aoms/1177705056