Summary
Let W(t) be a standard Wiener process and let f(x) be a function from the compact class in Strassen's law of the iterated logarithm. We investigate the lim inf behavior of the variable sup ¦W(xT)(2T loglog T)−1/2−f(x)¦, 0≦x≦1 suitably normalized as T→∞.
This extends Chung's result valid for f(x)≡0, stating that lim inf.[ sup ¦(2T loglogT)−1/2 W(xT)¦(loglog T)−1]=π/4 a.s. T→∞ 0≦x≦1
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Csáki, E. A relation between Chung's and Strassen's laws of the iterated logarithm. Z. Wahrscheinlichkeitstheorie verw Gebiete 54, 287–301 (1980). https://doi.org/10.1007/BF00534347
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DOI: https://doi.org/10.1007/BF00534347