Weak convergence of the empirical process and the rescaled empirical distribution function in the Skorokhod product space

TL;DR

This paper proves the asymptotic independence of the empirical process and a rescaled empirical distribution function in the Skorokhod space, revealing their joint convergence to independent limit processes.

Contribution

It establishes the weak convergence and asymptotic independence of these processes in the Skorokhod product space, extending classical convergence results to a more general setting.

Findings

The pair converges to a time-transformed Brownian bridge and a two-sided Poisson process.

Finite-dimensional convergence plus tightness implies weak convergence in the Skorokhod space.

Provides tightness criteria and generalizes known convergence results.

Abstract

We prove the asymptotic independence of the empirical process $\alpha_n = \sqrt{n}( F_n - F)$ and the rescaled empirical distribution function $\beta_n = n (F_n(\tau+\frac{\cdot}{n})-F_n(\tau))$, where $F$ is an arbitrary cdf, differentiable at some point $\tau$, and $F_n$ the corresponding empricial cdf. This seems rather counterintuitive, since, for every $n \in N$, there is a deterministic correspondence between $\alpha_n$ and $\beta_n$. Precisely, we show that the pair $(\alpha_n,\beta_n)$ converges in law to a limit having independent components, namely a time-transformed Brownian bridge and a two-sided Poisson process. Since these processes have jumps, in particular if $F$ itself has jumps, the Skorokhod product space $D(R) \times D(R)$ is the adequate choice for modeling this convergence in. We develop a short convergence theory for $D(R) \times D(R)$ by establishing the classical principle, devised by Yu. V. Prokhorov, that finite-dimensional convergence and tightness imply weak convergence. Several tightness criteria are given. Finally, the convergence of the pair $(\alpha_n,\beta_n)$ implies convergence of each of its components, thus, in passing, we provide a thorough proof of these known convergence results in a very general setting. In fact, the condition on $F$ to be differentiable in at least one point is only required for $\beta_n$ to converge and can be further weakened.