Komlós-Major-Tusnády approximations to increments of uniform empirical processes

The well-known Koml\ós-Major-Tusn\ády inequalities [Z. Wahrsch. Verw. Gebiete 32 (1975) 111-131; Z. Wahrsch. Verw. Gebiete 34 (1976) 33-58] provide sharp inequalities to partial sums of iid standard exponential random variables by a sequence of standard Brownian motions. In this paper, we employ these results to establish Gaussian approximations to weighted increments of uniform empirical and quantile processes. This approach provides rates to the approximations which, among others, have direct applications to statistics of extreme values for randomly censored data.