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, ... whereɛ 1,ɛ 2 ... are i.i.d. random variables,\(P\left( {\varepsilon _j = 1} \right) = P\left( {\varepsilon _j = - 1} \right) = \tfrac{1}{2}\) let α(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.
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Research supported by the Hungarian National Foundation for Scientific Research, Grant No # 819/1
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Major, P. On the set visited once by a random walk. Probab. Th. Rel. Fields 77, 117–128 (1988). https://doi.org/10.1007/BF01848134
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DOI: https://doi.org/10.1007/BF01848134