Rate of convergence in the central limit theorem for empirical processes
Let (\({\mathcal X},{\mathcal A},\mu)\) and (\({\mathcal X}^{{\mathbb{N}}},{\mathcal A}^{{\mathbb{N}}},\mu^{{\mathbb{N}}})\) be a probability space and its countable product, respectively. The empirical measure \(\mu_ n\) associated to \(\mu\) is defined by \(\mu_ n:=n^{-1}\sum^{n}_{i=1}\delta_{x_ i}\), \(n\in {\mathbb{N}}\), with \(\delta_ x\) denoting the point mass at \(x\in {\mathcal X}\). We set \(E_ n:=\sqrt{n}(\mu_ n-\mu),\) \(n\in {\mathbb{N}}\). \({\mathcal F}\) denotes a collection of real-valued measurable functions on (\({\mathcal X},{\mathcal A})\) such that \({\mathcal F}\subset {\mathcal L}^ 2({\mathcal X},{\mathcal A},\mu)\) and \[ (1)\quad F(x):=\sup_{f\in {\mathcal F}}| f(x)| <+\infty,\quad x\in {\mathcal X}. \] (\(\ell_{\infty}({\mathcal F}),\| \cdot \|_{{\mathcal F}})\) denotes the set of all bounded real- valued functions on \({\mathcal F}\) endowed with the sup norm \(\| \cdot \|_{{\mathcal F}}\). Let \(\mu (f):=\int f d\mu\) for any measure \(\mu\). Then the empirical processes indexed by \({\mathcal F}\) and based on \(\mu\) are defined by \(\{E_ n(f)\), \(f\in {\mathcal F}\}\), \(n\in {\mathbb{N}}.\) Let \({\mathcal F}\) be a countably determined class on (\({\mathcal X},{\mathcal A},\mu)\) satisfying (1). Put \[ \Delta_ n({\mathcal F},r):=| P(\{\| E_ n\|_{{\mathcal F}}>r\})-P(\{\| {\mathbb{B}}_{\mu}\|_{{\mathcal F}}<r\}),\quad \forall r\in {\mathbb{R}}_+,\quad n\in {\mathbb{N}})|, \] where \({\mathbb{B}}_{\mu}\) is a generalized Brownian bridge, \[ M_ 3(n):=n^{-1/6}([E \| \delta_ X-\mu \|^ 3_{{\mathcal F}}+E | {\mathbb{B}}_{\mu}\|^ 3_{{\mathcal F}})^{1/3}\vee 1),\quad n\in {\mathbb{N}}. \] The main result of this paper states that \[ \Delta_ n({\mathcal F},r)\leq C(1+r)^{-3}\{[1+H^ 2_ d(\delta_ n)][M_ 3(n)\vee M^ 3_ 3(n)]+\gamma_ n+\epsilon_ n]\},\quad \forall r\in {\mathbb{R}}_+, \] where \(\{\delta_ n\), \(n\in {\mathbb{N}}\}\), \(\{\gamma_ n\), \(n\in {\mathbb{N}}\}\), \(\{\epsilon_ n\), \(n\in {\mathbb{N}}\}\) are sequences of positive numbers decreasing to zero, and \(H_ d(u)\) denotes the metric entropy.
scientific article; zbMATH DE number 4215044
scientific article