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On the Behavior of Random Walk Around Heavy Points

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Abstract

Consider a symmetric aperiodic random walk in Z d, d≥3. There are points (called heavy points) where the number of visits by the random walk is close to its maximum. We investigate the local times around these heavy points and show that they converge to a deterministic limit as the number of steps tends to infinity.

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References

  1. Csáki, E., Földes, A., Révész, P.: Heavy points of a d-dimensional simple random walk. Stat. Probab. Lett. 76, 45–57 (2006)

    Article  MATH  Google Scholar 

  2. Csáki, E., Földes, A., Révész, P., Rosen, J., Shi, Z.: Frequently visited sets for random walks. Stoch. Process. Appl. 115, 1503–1517 (2005)

    Article  MATH  Google Scholar 

  3. Csáki, E., Földes, A., Révész, P.: Joint asymptotic behavior of local and occupation times of random walk in higher dimension. Studia Sci. Math. Hung. (2007, to appear)

  4. Erdős, P., Taylor, S.J.: Some problems concerning the structure of random walk paths. Acta Math. Acad. Sci. Hung. 11, 137–162 (1960)

    Article  Google Scholar 

  5. Erdős, P., Taylor, S.J.: Some intersection properties of random walk paths. Acta Math. Acad. Sci. Hung. 11, 231–248 (1960)

    Article  Google Scholar 

  6. Jain, N.C., Pruitt, W.E.: The range of transient random walk. J. Anal. Math. 24, 369–393 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  7. Révész, P.: The maximum of the local time of a transient random walk. Studia Sci. Math. Hung. 41, 379–390 (2004)

    MATH  Google Scholar 

  8. Révész, P.: Random Walk in Random and Non-random Environment, 2nd edn. World Scientific, Singapore (2005)

    Google Scholar 

  9. Spitzer, F.: Principles of Random Walk. Van Nostrand, Princeton (1964)

    MATH  Google Scholar 

  10. Uchiyama, K.: Green’s functions for random walks on Z N. Proc. Lond. Math. Soc. 77, 215–240 (1998)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, P.O.B. 127, 1364, Hungary

    Endre Csáki

  2. Department of Mathematics, College of Staten Island, CUNY, 2800 Victory Blvd., Staten Island, New York, 10314, USA

    Antónia Földes

  3. Institut für Statistik und Wahrscheinlichkeitstheorie, Technische Universität Wien, Wiedner Hauptstrasse 8-10/107, 1040, Vienna, Austria

    Pál Révész

Authors
  1. Endre Csáki
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  2. Antónia Földes
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  3. Pál Révész
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Corresponding author

Correspondence to Endre Csáki.

Additional information

Dedicated to Professor Wolfgang Wertz on his 60th birthday.

E. Csáki’s and P. Révész’s research supported by the Hungarian National Foundation for Scientific Research, Grant No. T 037886, T 043037 and K 061052.

A. Földes’ research supported by a PSC CUNY Grant, No. 66494-0035.

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Csáki, E., Földes, A. & Révész, P. On the Behavior of Random Walk Around Heavy Points. J Theor Probab 20, 1041–1057 (2007). https://doi.org/10.1007/s10959-007-0092-z

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  • DOI: https://doi.org/10.1007/s10959-007-0092-z

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