On the rate of convergence in Strassen's law of the iterated logarithm (Q1075679)

Let W(t) be a standard Wiener process and let \({\mathcal S}\) be the compact class figuring in Strassen's law of the iterated logarithm. We investigate the rate of convergence to zero of the variable \[ \inf_{f\in {\mathcal S}}\sup_{0\leq x\leq 1}| W(xT)(2T \log \log T)^{-1/2}-\quad f(x)|. \] It is shown that as \(T\to \infty\), (log log T)\({}^{-\alpha}\) belongs to the upper class of this variable if \(\alpha <2/3\), and to the lower class if \(\alpha >2/3\).