A CLT for weighted time-dependent uniform empirical processes

For a uniform process $\{ X_t: t\in E\}$ (by which $X_t$ is uniformly distributed on $(0,1)$ for $t\in E$) and a function $w(x)>0$ on $(0,1)$, we give a sufficient condition for the weak convergence of the empirical process based on $\{ w(x)(\mathbb{1}_{X_t\leq x} -x): t\in E, x\in [0,1]\}$ in $\ell^\infty(E\times [0,1])$. When specializing to $w(x)\equiv 1$ and assuming strict monotonicity on the marginal distribution functions of the input process, we recover a result of Kuelbs, Kurtz, and Zinn (2013). In the last section, we give an example of the main theorem.