The quantile process was shown by Bickel to converge in the uniform metric on intervals $\lbrack a, b\rbrack$ with $0 < a < b < 1$. By introducing appropriate new supremum metrics, this result is extended to all of (0, 1). Hence a natural process of ordered spacings from the uniform distribution converges in certain supremum metrics. This is used to establish the limiting normality of a large family of statistics based on ordered spacings, which can be used in testing for exponentiality. The non-null case is also considered.
Galen R. Shorack. "Convergence of Quantile and Spacings Processes with Applications." Ann. Math. Statist. 43 (5) 1400 - 1411, October, 1972. https://doi.org/10.1214/aoms/1177692373