A note on the rate of convergence in the CLT for empirical processes
Let \(X\), \(X_ 1\), \(X_ 2,\ldots\) be i.i.d. random variables with values in a measurable space \(({\mathcal X},{\mathcal A})\), and let \(\mu\) be the distribution of \(X\). The empirical measure \(\mu_ n\) associated with \(\mu\) and the empirical process \(E_ n\) are defined by the formulas \[ \mu_ n:=n^{-1}\sum^ n_{i=1}\delta_{x_ i},\qquad E_ n:=\sqrt n(\mu_ n-\mu). \] We consider the empirical process indexed by a class \({\mathcal F}\) of real-valued measurable functions on \({\mathcal X}\) such that \({\mathcal F}\subset L_ 2({\mathcal X},{\mathcal A},\mu)\) and \(F(x):=\sup_{f\in{\mathcal F}}| f(x)|<\infty\) \(\mu\)-a.e. For \(f, g\in{\mathcal F}\) and for any measure \(\nu\) on \(({\mathcal X},{\mathcal A})\) we use the notation \[ \nu(f)=\int f(x)\nu(dx),\quad \|\nu\|_{{\mathcal F}}=\sup_{f\in{\mathcal F}}\mid\nu(f)|, \] \[ e_{\mu,p}(f,g)= \begin{cases} (\mu(| f-g|^ p))^{1/p},\quad &1\leq p<\infty, \\ \mu- \text{ess}\sup| f-g|,\quad &p=\infty.\end{cases} \] Suppose that \(\mu\) and the class \({\mathcal F}\) satisfy the central limit theorem [see, for example, \textit{R. M. Dudley}, Ann. Probab. 6, 899-929 (1978; Zbl 0404.60016)] and denote a limiting Gaussian process by \(\{G_ \mu(f), f\in{\mathcal F}\}\). Then one possible way of estimating the rate of convergence in this theorem is to obtain bounds for the quantities \(\Delta_ n({\mathcal F})=\sup_{r\geq 0}\Delta_ n({\mathcal F},r)\), where \(\Delta_ n({\mathcal F},r)=| P\{\| E_ n\|_{\mathcal F}>r\}- P\{\| G_ \mu\|_{\mathcal F}>r\}|\).
scientific article; zbMATH DE number 578658