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Thick points for planar Brownian motion and the Erdős-Taylor conjecture on random walk

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References

  1. Bass, R. F. &Griffin, P. S., The most visited site of Brownian motion and simple random walk.Z. Wahrsch. Verw. Gebiete, 70 (1985), 417–436.

    Article  MathSciNet  Google Scholar 

  2. Bingham, N. H., Goldie, C. M. &Teugels, J. L.,Regular Variation. Encyclopedia Math. Appl., 27. Cambridge Univ. Press., Cambridge, 1987.

    Google Scholar 

  3. Dembo, A., Peres, Y., Rosen, J. &Zeitouni, O., Thick points for spatial Brownian motion: multifractal analysis of occupation measure.Ann. Probab., 28 (2000), 1–35.

    MathSciNet  Google Scholar 

  4. —, Thin points for Brownian motion.Ann. Inst. H. Poincaré Probab. Statist., 36 (2000), 749–774.

    Article  MathSciNet  Google Scholar 

  5. Dembo, A. &Zeitouni, O.,Large Deviations Techniques and Applications, 2nd edition. Appl. Math., 38. Springer-Verlag, New York, 1998.

    Google Scholar 

  6. Einmahl, U., Extensions of results of Komlós, Major, and Tusnády to the multivariate case.J. Multivariate Anal., 28 (1989), 20–68.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Kahane, J.-P.,Some Random Series of Functions, 2nd edition. Cambridge Stud. Adv. Math., 5. Cambridge Univ. Press, Cambridge, 1985.

    Google Scholar 

  9. Kaufman, R., Une propriété metriqué du mouvement brownien.C. R. Acad. Sci. Paris Sér. A-B, 268 (1969), A727-A728.

    MathSciNet  Google Scholar 

  10. Komlós, J., Major, P. &Tusnády, G., An approximation of partial sums of independent RV's, and the sample DF, I.Z. Wahrsch. Verw. Gebiete, 32 (1975), 111–131.

    Article  Google Scholar 

  11. Le Gall, J.-F. &Rosen, J., The range of stable random walks.Ann. Probab., 19 (1991), 650–705.

    MathSciNet  Google Scholar 

  12. Lyons, R. &Pemantle, R., Random walks in a random environment and first-passage percolation on trees.Ann. Probab., 20 (1992), 125–136.

    MathSciNet  Google Scholar 

  13. Mattila, P.,Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability. Cambridge Stud. Adv. Math., 44. Cambridge Univ. Press, Cambridge, 1995.

    Google Scholar 

  14. Mörters, P., The average density of the path of planar Brownian motion.Stochastic Process. Appl., 74 (1998), 133–149.

    Article  MATH  MathSciNet  Google Scholar 

  15. Orey, S. &Taylor, S. J., How often on a Brownian path does the law of the iterated logarithm fail?Proc. London Math. Soc. (3), 28 (1974), 174–192.

    MathSciNet  Google Scholar 

  16. Pemantle, R., Peres, Y., Pitman, J. & Yor, M., Where did the Brownian particle go? To appear inElectron. J. Probab.

  17. Perkins, E. A. &Taylor, S. J., Uniform measure results for the image of subsets under Brownian motion.Probab. Theory Related Fields, 76 (1987), 257–289.

    Article  MathSciNet  Google Scholar 

  18. Ray, D., Sojourn times and the exact Hausdorff measure of the sample path for planar Brownian motion.Trans. Amer. Math. Soc., 106 (1963), 436–444.

    Article  MathSciNet  Google Scholar 

  19. Révész, P.,Random Walk in Random and Non-Random Environments. World Sci. Publishing, Teaneck, NJ, 1990.

    Google Scholar 

  20. Spitzer, F.,Principles of Random Walk. Van Nostrand, Princeton, NJ, 1964.

    Google Scholar 

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The first author was partially supported by NSF Grant DMS-9704552-002; the second author was partially supported by NSF Grant DMS-9803597; the third author was supported in part by grants from the NSF and from PSC-CUNY; the work of all authors was supported in part by a US-Israel BSF grant.

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Dembo, A., Peres, Y., Rosen, J. et al. Thick points for planar Brownian motion and the Erdős-Taylor conjecture on random walk. Acta Math. 186, 239–270 (2001). https://doi.org/10.1007/BF02401841

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