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

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We prove the asymptotic independence of the empirical process αn = √n(Fn -F) and the rescaled empirical distribution function βn = n(Fn(τ + ̇/n )-Fn(τ)), where F is an arbitrary characteristic distribution function, differentiable at some point τ, and Fn is the corresponding empirical characteristic distribution function. This seems rather counterintuitive, since, for every n ε N, there is a deterministic correspondence between αn and βn. Precisely, we show that the pair (αn, βn) 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)×D(R) is the adequate choice for modeling this convergence in. We develop a short convergence theory for D(R) × D(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 (αn, βn) 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 βn 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



  • Brownian bridge, Finite-dimensional distribution, Poisson process, Skorokhod topology, Tightness

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