A refinement of the Komlos, Major and Tusnady (1975) inequality for the supremum distance between the uniform empirical process and a constructed sequence of Brownian bridges is obtained. This inequality leads to a weighted approximation of the uniform empirical and quantile processes by a sequence of Brownian bridges dual to that recently given by M. Csorgo, S. Csorgo, Horvath and Mason (1986). The present theory approximates the uniform empirical process more closely than the uniform quantile process, whereas the former theory more closely approximates the uniform quantile process.
David M. Mason. Willem R. Van Zwet. "A Refinement of the KMT Inequality for the Uniform Empirical Process." Ann. Probab. 15 (3) 871 - 884, July, 1987. https://doi.org/10.1214/aop/1176992070