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Three favorite sites occurs infinitely often for one-dimensional simple random walk. (English) Zbl 1428.60060
Summary: For a one-dimensional simple random walk \((S_{t})\), for each time \(t\) we say a site \(x\) is a favorite site if it has the maximal local time. In this paper, we show that with probability 1 three favorite sites occurs infinitely often. Our work is inspired by B. Tóth [Ann. Probab. 29, No. 1, 484–503 (2001; Zbl 1031.60036)], and disproves a conjecture of P. Erdős and P. Révész [in: Mathematical statistics and probability theory, Proc. 6th Pannonian Symp., Bad Tatzmannsdorf/Austria 1986, Vol. B, 59–65 (1987; Zbl 0629.60081)] and of Tóth [loc. cit.].

MSC:
60G50 Sums of independent random variables; random walks
60J55 Local time and additive functionals
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