Precise asymptotics on second-order complete moment convergence of uniform empirical process (Q1722176)

Summary: Let \(\{\xi_i, 1 \leq i \leq n \}\) be a sequence of iid \(U[0, 1]\)-distributed random variables, and define the uniform empirical process \(F_n(t) = n^{- 1 / 2} \sum_{i = 1}^n (I_{\{\xi_i \leq t \}} - t)\), \(0 \leq t \leq 1\), \(\|F_n\| = \sup_{0 \leq t \leq 1} | F_n(t) |\). When the nonnegative function \(g(x)\) satisfies some regular monotone conditions, it proves that \(\lim_{\varepsilon \searrow 0} \left(1 / - \log \varepsilon\right) \sum_{n = 1}^{\infty} \left(g'(n) / g(n)\right) \mathbb{E} \{\|F_n\|^2 I_{\{\| F_n \|l \geq \varepsilon \sqrt{g(n)} \}} \} = \pi^2 / 6\).