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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
##### Keywords:
Wiener process; large deviation
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