An elementary proof of the weak convergence of empirical processes

Marten Wegkamp
Department of Mathematics and Department of Statistical Sciences
Cornell University

In this joint work with Dragan Radulovic, we develop a simple technique for proving the weak convergence of a stochastic process 
$\bar{\mathbb{Z}}_{n}(g):=\int g\,\mathrm{d}\mathbb{Z}_{n}$, indexed by functions $g$ on the real line in some  class $\mathcal{G}$.
The main novelty is a decoupling argument that allows to derive asymptotic equicontinuity of the process $\{ \bar{\mathbb{Z}}_{n}(g)$, $g\in\G\}$
from that of the basic process $\{\ZZ_n(t), \ t\in\RR\}$, with $\ZZ_n(t)= \bar{\ZZ}_n(f_t)$ and $f_t(x)=1_{(-\infty,t]}(x)$.
The method leads to novel results for empirical processes based on stationary processes and its bootstrap versions.
If time permits, we discuss extensions to weak convergence of the empirical copula process indexed by functions with bounded variation on the unit cube.