A CLT for weighted time-dependent uniform empirical processes

For a uniform process { X t : t ∈ E } (by which X t is uniformly distributed on ( 0 , 1 ) for t ∈ E ) and a function w ( x ) > 0 on ( 0 , 1 ) , we give a sufficient condition for the weak convergence of the empirical process based on { w ( x ) ( 1 X t ≤ x − x ) : t ∈ E , x ∈ [ 0 , 1 ] } in ℓ ∞ ( E × [ 0 , 1 ] ) . When specializing to w ( 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.