References
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.
Bingham, N. H., Goldie, C. M. &Teugels, J. L.,Regular Variation. Encyclopedia Math. Appl., 27. Cambridge Univ. Press., Cambridge, 1987.
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.
—, Thin points for Brownian motion.Ann. Inst. H. Poincaré Probab. Statist., 36 (2000), 749–774.
Dembo, A. &Zeitouni, O.,Large Deviations Techniques and Applications, 2nd edition. Appl. Math., 38. Springer-Verlag, New York, 1998.
Einmahl, U., Extensions of results of Komlós, Major, and Tusnády to the multivariate case.J. Multivariate Anal., 28 (1989), 20–68.
Erdős, P. &Taylor, S. J., Some problems concerning the structure of random walk paths.Acta Math. Acad. Sci. Hungar., 11 (1960), 137–162.
Kahane, J.-P.,Some Random Series of Functions, 2nd edition. Cambridge Stud. Adv. Math., 5. Cambridge Univ. Press, Cambridge, 1985.
Kaufman, R., Une propriété metriqué du mouvement brownien.C. R. Acad. Sci. Paris Sér. A-B, 268 (1969), A727-A728.
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.
Le Gall, J.-F. &Rosen, J., The range of stable random walks.Ann. Probab., 19 (1991), 650–705.
Lyons, R. &Pemantle, R., Random walks in a random environment and first-passage percolation on trees.Ann. Probab., 20 (1992), 125–136.
Mattila, P.,Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability. Cambridge Stud. Adv. Math., 44. Cambridge Univ. Press, Cambridge, 1995.
Mörters, P., The average density of the path of planar Brownian motion.Stochastic Process. Appl., 74 (1998), 133–149.
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.
Pemantle, R., Peres, Y., Pitman, J. & Yor, M., Where did the Brownian particle go? To appear inElectron. J. Probab.
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.
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.
Révész, P.,Random Walk in Random and Non-Random Environments. World Sci. Publishing, Teaneck, NJ, 1990.
Spitzer, F.,Principles of Random Walk. Van Nostrand, Princeton, NJ, 1964.
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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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DOI: https://doi.org/10.1007/BF02401841