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 Rényi 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 \(0\leq\nu<1/4\) in the empirical case and for \(0\leq\eta<1/2\) 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.