Doob  has given heuristically an appealing methodology for deriving asymptotic theorems on the difference between the empirical distribution function calculated from a sample and the actual distribution function of the population being sampled. In particular he has applied these methods to deriving the well known theorems of Kolmogorov  and Smirnov . In this paper we give a justification of Doob's approach to these theorems and show that the method can be extended to a wide class of such asymptotic theorems.
"Justification and Extension of Doob's Heuristic Approach to the Kolmogorov- Smirnov Theorems." Ann. Math. Statist. 23 (2) 277 - 281, June, 1952. https://doi.org/10.1214/aoms/1177729445