Weighted Gaussian Approximations for Increments of the Uniform Empirical and Quantile Processes: Fixed-Endpoint Extensions to the Finite-Count Scale

We establish weighted Gaussian approximations for the uniform empirical and quantile processes and for their increments ending at a fixed point t in (0,1). We first place the classical weighted approximations for the ordinary processes in a common framework and then show that the corresponding increment approximations remain valid uniformly down to the finite-count scale lambda/n, for every fixed lambda > 0. For the empirical increments, the proof splits the sample at t, couples the two resulting conditional empirical processes with independent Brownian bridges, and approximates the binomial fluctuation at t by a Gaussian variable. The three Gaussian components are then combined into a single standard Brownian bridge. For the quantile increments, the Renyi representation and a reversal of the relevant exponential spacings reduce the problem to the weighted approximation of an ordinary uniform quantile process. The resulting bounds hold for 0 <= nu < 1/4 in the empirical case and for 0 <= eta < 1/2 in the quantile case. As an application of the empirical increment approximation, we derive simultaneous weighted Gaussian approximations for the censored and uncensored empirical subdistribution-tail processes arising under random right censoring.