Locations of extreme values of the empirical process (Q2772065)

Let \(\alpha_n\) be a uniform empirical process and \(\mu_n\) (respectively, \(\nu_n)\) the unique location of its maximum (respectively, minimum). The author proves that NEWLINE\[NEWLINE\liminf_{n\to\infty} \log^2_2(n) |\mu_n-\nu_n |=\pi^2 \text{ almost surely}.NEWLINE\]NEWLINE The basic tool of proof is a strong approximation theorem due to \textit{J. Komlós}, \textit{P. Major} and \textit{G. Tusnády} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 32, 111-131 (1975; Zbl 0308.60029)].