Let W(t), \(0\leq t<\infty\), be a Brownian motion. This paper shows that \[ \limsup_{T\to \infty}\sup_{0<t\leq T}\frac{| W(T)-W(T- t)|}{\{2t(\log \quad (T/t)+\log \log t)\}^{1/2}}=1,\quad a.s. \]
\[ \lim_{T\to \infty}\sup_{0<t\leq T}\sup_{t\leq s\leq T}\frac{| W(T)-W(s-t)|}{\{2t(\log (T/t)+\log \log t)\}^{1/2}}=1,\quad a.s. \] These results give an affirmative answer to the questions posed by D. L. Hanson and R. P. Russo, Ann. Probab. 11, 609-623 and 1009- 1015 (1983; Zbl 0519.60030 and Zbl 0521.60033, respectively).
These results and their corollaries imply many important results about lim sup sup properties of increments of Brownian motion, such as the law of iterated logarithms for Brownian motion, the M. Csörgö and P. Révész theorem [ibid. 7, 731-737 (1979; Zbl 0412.60038)], and Hanson-Russo theorem (op. cit.).
The details of the proofs will be published in Ann. Probab. 14 (1986).