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 -