Exact rate of convergence in Strassen's law of the iterated logarithm (Q1185803)

Let be \(\eta_ T(t)=W(Tt)/\sqrt{2T \log \log T}\), where \(W(t)\) is a standard Wiener process on \((\Omega,{\mathcal F},P)\), \(t\in[0,1]\), and \({\mathcal S}=\{f(t)=\int_ 0^ t g(u)du:\int_ 0^ 1 g^ 2(u)du\leq 1\}\). The author proves that \(0<\limsup_{T\to\infty}(\log \log T)^{2/3}d(T)<\infty\), where \(d(T)=\inf\{\|\eta_ T-f\|_ \infty:\;f\in{\mathcal S}\}\).