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

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.

References in corpus (1)

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

Cited by in corpus (2)

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