An elementary proof of the weak convergence of empirical processes

This paper develops a simple technique for proving the weak convergence of a stochastic process Z̄n(g) := ∫ g dZn, indexed by functions g in some class G. The main novelty is a decoupling argument that allows to derive asymptotic equicontinuity of the process {Z̄n(g), g ∈ G} from that of the basic process {Zn(t), t ∈ R}, with Zn(t) = Z̄n(ft) and ft(x) = 1(−∞,t](x). The method leads to novel results for empirical processes based on stationary processes and its bootstrap versions. Running title: Weak convergence of empirical processes indexed by functions of bounded variation. MSC2000 Subject classification: Primary 60F17 ; secondary 60G99.