The uniform empirical process $U_n$ is considered as a process indexed by intervals. Powerful new exponential bounds are established for the process indexed by both "points" and intervals. These bounds trivialize the proof of the Chibisov-O'Reilly theorem concerning the convergence of the process with respect to $\|\cdot/q\|$-metrics and are used to prove an interval analogue of the Chibisov-O'Reilly theorem. A strong limit theorem related to the well-known Holder condition for Brownian bridge $U$ is also proved. Connections with related work of Csaki, Eicker, Jaeschke, and Stute are mentioned. As an application we introduce a new statistic for testing uniformity which is the natural interval analogue of the classical Anderson-Darling statistic.
"Limit Theorems and Inequalities for the Uniform Empirical Process Indexed by Intervals." Ann. Probab. 10 (3) 639 - 652, August, 1982. https://doi.org/10.1214/aop/1176993773