Inner rates of coverage of Strassen type sets by increments of the uniform empirical and quantile processes (Q2485838)

The author studies, among others, the strong limiting behavior of the empirical oscillation modulus at \(f\in B\) where \(B\) denotes the Banach space of all bounded real-valued functions defined on \([0, 1]\) and vanishing at \(0\), endowed with the sup-norm \(\|\cdot\|\). More specifically, let \(\{U_i; i\geq 1\}\) be i.i.d. uniform \((0, 1]\) random variables and define the empirical increment process \[ \Delta\alpha_n(a,t)= \alpha_n(t+ aI)- \alpha_n(t)\in B,\quad a\in [0,1],\quad t\in [0,1-a], \] where \(\alpha_n\) denotes the empirical process based on \(U_1,\dots, U_n\). Define \(J(f)= \int_{[0,1]} (f')^2d\lambda\) and let \(\{a_n\}\) be a non-random sequence such that \(a_n\downarrow 0\) and \(na_n\uparrow\infty\). Put \(c= \lim_{n\to\infty}\, (\log a^{-1}_n)/(\log\log n).\) Assuming that \(f\in\{g: J(g)< 1\}\) and some additional conditions the author proves that if \(c=\infty\), then \[ \lim_{n\to\infty}\,\nabla_f(\log a^{-1}_n)\inf_{t\in T_n}\,\Biggl\|{\Delta\alpha_n(a_n, t)\over b_n}- f\Biggr\|= {\pi\over 4\sqrt{1-J(f)}}, \] and that if \(c< \infty\), then \[ \liminf_{n\to\infty}\, \nabla_f((1+ c)\log\log n)\inf_{t\in T_n}\,\Biggl\|{\Delta\alpha_n(a_n, t)\over b_n}- f\Biggr\|= {\pi\over 4\sqrt{1-J(f)}}, \] where \(b_n=n(2a_n \log a^{-1}_n+ \log\log n)^{1/2}\), \(T_n\) is a subset of \([0,1-a_n]\) and \(\nabla_f(L)\) is the solution of the equation \(1- \inf_{\| g-f\|\leq 1/\nabla} J(g)= \pi^2\nabla^2/16L^2.\)