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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2 Citations (Scopus)

Abstract


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
Volume54
Issue number4
DOIs
Publication statusPublished - 2010

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Empirical Distribution Function
Empirical Process
Product Space
Weak Convergence
Tightness
Characteristic Function
Differentiable
Distribution Function
Asymptotic Independence
Converge
Imply
Brownian Bridge
Jump Process
Convergence Theory
Poisson process
Convergence Results
Jump
Correspondence
Arbitrary
Modeling

Cite this

@article{265916911c0646e0b51d680ac4085f93,
title = "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 $\alpha_n = \sqrt{n}( {\mathbb F}_n - F)$ 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.",
author = "Daniel Vogel and Dietmar Ferger",
year = "2010",
doi = "10.1137/S0040585X97984486",
language = "English",
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journal = "Theory of Probability and its Applications",
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TY - JOUR

T1 - Weak convergence of the empirical process and the rescaled empirical distribution function in the Skorokhod product space

AU - Vogel, Daniel

AU - Ferger, Dietmar

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

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DO - 10.1137/S0040585X97984486

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