A CLT for weighted time-dependent uniform empirical processes

Yuping Yang

doi:10.1016/j.jspi.2014.09.002

paper

A CLT for weighted time-dependent uniform empirical processes

arXiv:1412.8162 · doi:10.1016/j.jspi.2014.09.002

Abstract

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 ${w (x) (1_{X_{t} \leq x} - x) : t \in E, x \in [0, 1]}$ in $ℓ^{\infty} (E \times [0, 1])$ . 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.