On the Speed of Convergence in Strassen's Law of the Iterated Logarithm

Here there is derived a condition on sequences $\varepsilon_n \downarrow 0$ which implies that $P\lbrack W(n^\bullet)/(2n \log \log n)^\frac{1}{2} \not\in K^\varepsilon n \mathrm{i.o.}\rbrack = 0$, where $W$ is the Wiener process and $K$ is the compact set in Strassen's law of the iterated logarithm. A similar result for random walks is also given.