Chung's law and the Csáki function (Q1295854)

The solution to the problem of convergence rate in Chung's functional law of the iterated logarithm is obtained for so-called ``slowest points'' \(h \in C[0,1]\). Namely, the authors find the value \[ {\mathcal L}_h = \liminf_{T \to \infty} (\log\log T)^{2/3} \Biggl\|\frac{W(T\cdot)}{(2T\log\log T)^{1/2}} - h\Biggr\| \] where \(W\) is a standard Wiener process, \(||\cdot||\) is the sup-norm in the space \(C[0,1]\). This is a generalization of results of \textit{E. Csáki} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 54, 287-301 (1980; Zbl 0441.60027) and in: Limit theorems in probability and statistics. Colloq. Math. Soc. J. Bolyai 57, 83-93 (1990; Zbl 0719.60033)] proved both for piecewise-linear function \(h\) and quadratic function \(h\). The problem of finding \({\mathcal L}_h\) is related to the study of some functional of Wiener process in a strip.