A CLT for weighted time-dependent uniform empirical processes

•Found a sufficient condition for time-dependent weighted empirical processes.•Give an example for iid samples of Brownian motion for a class of functions of weights.•Proved the sample boundedness and continuity property by comparison of L2L2 distances.

For a uniform process {Xt:t∈E}{Xt:t∈E} (by which XtXt is uniformly distributed on (0,1)(0,1) for t∈Et∈E) and a function w(x)>0w(x)>0 on (0,1)(0,1), we give a sufficient condition for the weak convergence of the empirical process based on {w(x)(1Xt≤x−x):t∈E,x∈[0,1]}{w(x)(1Xt≤x−x):t∈E,x∈[0,1]} in ℓ∞(E×[0,1])ℓ∞(E×[0,1]). When specializing to w(x)≡1w(x)≡1 and assuming strict monotonicity on the marginal distribution functions of the input process, we recover a result of Kuelbs et al. (2013). In the last section, we give an example of the main theorem.