### 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 language | English |
---|---|

Pages (from-to) | 609-625 |

Number of pages | 17 |

Journal | Theory of Probability and its Applications |

Volume | 54 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2010 |

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### Cite this

**Weak convergence of the empirical process and the rescaled empirical distribution function in the Skorokhod product space.** / Vogel, Daniel; Ferger, Dietmar.

Research output: Contribution to journal › Article

*Theory of Probability and its Applications*, vol. 54, no. 4, pp. 609-625. https://doi.org/10.1137/S0040585X97984486

}

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.

U2 - 10.1137/S0040585X97984486

DO - 10.1137/S0040585X97984486

M3 - Article

VL - 54

SP - 609

EP - 625

JO - Theory of Probability and its Applications

JF - Theory of Probability and its Applications

IS - 4

ER -