Locating the maximum of an empirical process (Q1359722)

Let \(\alpha_n\) be a uniform empirical process, \(\xi_n\) the unique location of its absolute maximum realized at an order statistic point, and let \(\rho_n\) be the index of the order statistic point where \(\alpha_n\) realizes its absolute maximum. Then the following iterated logarithm laws hold almost surely: \[ \lim\inf (\log\log n)^2 \xi_n= \pi^2/4, \quad \lim\inf (\log \log n)^2 (1-\xi_n) =\pi^2/4, \] \[ \lim\inf (\log\log n)^2 \rho_n/n= \pi^2/4,\quad \lim\inf (\log \log n)^2(n-\rho_n)/n =\pi^2/4. \]