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On the set visited once by a random walk

Authors:	Péter Major

Institution:	(1) Mathematical Institute of the Hungarian Academy of Sciences, Reáltanoda u. 13-15, H-1053 Budapest, Hungary

Abstract:	Summary In this paper we prove the following statement. Given a random walk $S_n = \sum\limits_{j = 1}^n {\varepsilon _j }$ ,n=1, 2, ... where ₁, ₂ ... are i.i.d. random variables, $P\left( {\varepsilon _j = 1} \right) = P\left( {\varepsilon _j = - 1} \right) = \tfrac{1}{2}$ let (n) denote the number of points visited exactly once by this random walk up to timen. We show that there exists some constantC, 0 <C < , such that $\mathop {\lim \sup }\limits_{n \to \infty } \frac{{\alpha (n)}}{{\log ^2 n}} = C$ with probability 1. The proof applies some arguments analogous to the techniques of the large deviation theory.Research supported by the Hungarian National Foundation for Scientific Research, Grant N^o # 819/1

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