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

Daniel Vogel, Dietmar Ferger

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We prove the asymptotic independence of the empirical process $\alpha_n = \sqrt{n}( {\mathbb F}_n - F)$ and the rescaled empirical distribution function $\beta_n = n ({\mathbb F}_n(\tau+\frac{\cdot}{n})-{\mathbb F}_n(\tau))$, where F is an arbitrary characteristic distribution function, differentiable at some point $\tau$, and ${\mathbb F}_n$ is the corresponding empirical characteristic distribution function. This seems rather counterintuitive, since, for every $n \in {\bf N}$, 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 $D({\bf R}) \times D({\bf R})$ is the adequate choice for modeling this convergence in. We develop a short convergence theory for $D({\bf R}) \times D({\bf R})$ by establishing the classical principle, devised by Yu. 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.
Original languageEnglish
Pages (from-to)609-625
Number of pages17
JournalTheory of Probability and its Applications
Issue number4
Publication statusPublished - 2010


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