Answers to some questions about increments of a Wiener process
The following theorems are proved: 1. Let W(t), \(0\leq t<\infty\), be a standard Wiener process. We have \[ \limsup_{T\to \infty}\sup_{0<t\leq T}\frac{| W(t)-W(T- t)|}{\{2t(\log (T/t)+\log \log t)\}^{1/2}}=1,\quad a.s., \] \[ \lim_{T\to \infty}\sup_{0<t\leq T}\sup_{t\leq s\leq T}\frac{| W(s)-W(s-t)|}{\{2t(\log (T/t)+\log \log t)\}^{1/2}}=1,\quad a.s., \] \[ \limsup_{T\to \infty}\sup_{0<t\leq T}\sup_{0\leq s\leq t}\frac{| W(T)-W(T-s)|}{\{2t(\log (T/t)+\log \log t)\}^{1/2}}=1,\quad a.s.. \] 2. Under the conditions (a) \(a_ T\to \infty\) continuously as \(T\to \infty\), (b) \(\lim_{T\to \infty}\log (T/a_ T)/\log \log a_ T=\infty:\) \[ \lim_{t\to \infty}\sup_{0\leq t\leq T-a_ T}\frac{| W(t+a_ T)-W(t)|}{\{2a_ T(\log ((t+a_ T)/a_ T)+\log \log a_ T)\}^{1/2}}=1,\quad a.s., \] \[ \lim_{T\to \infty}\sup_{0\leq t\leq T-a_ T}\sup_{0\leq s\leq a_ T}\frac{| W(t+s)-W(t)|}{\{2a_ T(\log ((t+a_ T)/a_ T)+\log \log a_ T)\}^{1/2}}=1,\quad a.s.. \] The results of the paper answer the questions posed by \textit{D. L. Hanson} and \textit{R. P. Russo} [ibid. 11, 609-623 (1983; Zbl 0519.60030)] and imply some results by \textit{M. Csörgö} and \textit{P. Révész} [ibid. 7, 731-737 (1979; Zbl 0412.60038) and Strong approximations in probability and statistics. (1981; Zbl 0539.60029)].
convergence properties of increments of a Wiener process
scientific article; zbMATH DE number 3992536