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On the set visited once by a random walk
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  • Published: March 1988

On the set visited once by a random walk

  • Péter Major1 

Probability Theory and Related Fields volume 77, pages 117–128 (1988)Cite this article

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  • 2 Citations

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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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References

  1. Csáki, E., Mohanty, S.G.: Excursion and meander in random walk. Can. J. Stat.9, 57–70 (1981)

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  2. Csáki, E., Mohanty, S.G.: Some joint distributions for conditional random walks. Can. J. Stat.14, 19–28 (1986)

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  3. Feller, W.: An introduction to probability theory and its applications, vol. 1. New York Chichester Brisbane Toronto: Wiley 1970

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Author information

Authors and Affiliations

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

    Péter Major

Authors
  1. Péter Major
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Additional information

Research supported by the Hungarian National Foundation for Scientific Research, Grant No # 819/1

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Cite this article

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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  • Received: 01 October 1986

  • Revised: 31 August 1987

  • Issue Date: March 1988

  • DOI: https://doi.org/10.1007/BF01848134

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Keywords

  • Stochastic Process
  • Random Walk
  • Probability Theory
  • Mathematical Biology
  • Deviation Theory
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