Precise Asymptotics on Second-Order Complete Moment Convergence of Uniform Empirical Process

Abstract

Let $\{{\xi }_i,1\le i\le n\}$ be a sequence of iid U[0, 1]-distributed random variables, and define the uniform empirical process $F_n(t)=n^{-1/2}{\sum }_{i=1}^n\mathrm{‍}(I_{\{{\xi }_i\le t\}}-t),0\le t\le 1$, $∥F_n∥={\text{s}\text{u}\text{p}}_{0\le t\le 1}|F_n(t)|$. When the nonnegative function $g(x)$ satisfies some regular monotone conditions, it proves that ${\text{lim}}_{ϵ\searrow 0}\left(1/-\text{l}\text{o}\text{g}ϵ\right){\sum }_{n=1}^{\mathrm{\infty }}\left(g^{\mathrm{\prime }}(n)/g(n)\right)\mathbb{E}\{{∥F_n∥}^2{I}_{\{\parallel F_n\parallel \ge ϵ\sqrt{g(n)}\}}\}={\pi }^2/6$.