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Thick points for planar Brownian motion and the Erdős-Taylor conjecture on random walk. (English) Zbl 1008.60063
P. Erdős and S. J. Taylor [Acta Math. Acad. Sci. Hung. 11, 137-162 (1960; Zbl 0091.13303)] proved that both the lim inf and the lim sup of \(T^*_n (\log n)^{-1}\) lie a.s. between \(1/4\pi\) and \(1/ \pi\), where \(T^*_n: =\max_{x\in Z^2} \sum^n_{j=1} I[X_j=x]\), and \(X\) is simple planar random walk. In this paper, their conjecture that both are a.s. equal to \(1/ \pi\) is proved, together with a number of extensions and refinements. The method depends on a multiscale adaptation of the classical second moment method which is used to prove analogous results for the occupation measure of planar Brownian motion. These are extended to the random walk setting by using strong approximation theorems as by J. Komlós, P. Major and G. Tusnády [Z. Wahrscheinlichkeitstheorie Verw. Geb. 32, 111-131 (1975; Zbl 0308.60029)] and U. Einmahl [J. Multivariate Anal. 28, No. 1, 20-68 (1989; Zbl 0676.60038)]. Similar results for a larger class of random walks are also given.

60G50 Sums of independent random variables; random walks
60J65 Brownian motion
60F15 Strong limit theorems
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