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Chung’s law and the Csáki function. (English) Zbl 0937.60013
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 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.

MSC:
60F15 Strong limit theorems
60G15 Gaussian processes
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