Abstract
We survey the current status of the list of questions related to the favourite (or: most visited) sites of simple random walk on Z, raised by Pál Erdős and Pál Révész in the early eighties.
Similar content being viewed by others
REFERENCES
R. F. Bass, N. Eisenbaum and Z. Shi, The most visited sites of symmetric stable processes, Probab. Th. Rel. Fields, 116 (2000), 391–404.
R. F. Bass and P. S. Griffin, The most visited site of Brownian motion and simple random walk, Z. Wahrsch. Verw. Gebiete, 70 (1985), 417–436.
J. Bertoin and L. Marsalle, Point le plus visité par un mouvement brownien avec dérive, Sém. Probab. XXXII, Lecture Notes in Mathematics, 1686, pp. 397–411. Springer, Berlin, 1998.
A. N. Borodin, Distributions of functionals of Brownian local time. II, Theory Probab. Appl. 34 (1989), 576–590.
E. Csáki, P. Révész and Z. Shi, Favourite sites, favourite values and jump sizes for random walk and Brownian motion, Bernoulli, to appear.
E. Csáki and Z. Shi, Large favourite sites of simple random walk and the Wiener process, Electronic J. Probab. 3 (1998), paper no. 14, pp. 1–31.
E. Csáki and Z. Shi, In preparation.
A. Dembo, Y. Peres, J. Rosen and O. Zeitouni, Thick points for planar Brownian motion and the Erdős–Taylor conjecture on random walk, Acta Math., to appear.
N. Eisenbaum, On the most visited sites by a symmetric stable process, Probab. Th. Rel. Fields, 107 (1997), 527–535.
N. Eisenbaum and D. Khoshnevisan, On the most visited sites of symmetric Markov processes, in preparation.
P. Erdős and P. Révész, On the favourite points of a random walk, Mathematical Structures–Computational Mathematics–Mathematical Modelling, 2 (1984), pp. 152–157. Sofia.
P. Erdős and P. Révész, Problems and results on random walks, In: Mathematical Statistics and Probability (P. Bauer et al., eds.), Proceedings of the 6th Pannonian Symposium, Vol. B, pp. 59–65. Reidel, Dordrecht, 1987.
Y. Hu and Z. Shi, Favourite sites of transient Brownian motion, Stoch. Proc. Appl. 73 (1998), 87–99.
Y. Hu and Z. Shi, The problem of the most visited site in random environment, Probab. Th. Rel. Fields, 116 (2000), 273–302.
H. Kesten, An iterated logarithm law for local time, Duke Math. J. 32 (1965), 447–456.
D. Khoshnevisan and T. M. Lewis, The favorite point of a Poisson process, Stoch. Proc. Appl. 57 (1995), 19–38.
C. Leuridan, Le point d'un fermé le plus visité par le mouvement brownien, Ann. Probab. 25 (1997), 953–996.
P. Major, On the set visited once by a random walk, Probab. Th. Rel. Fields, 77 (1988), 117–128.
M. B. Marcus, The most visited sites of certain Lévy processes, preprint.
P. Révész, Local time and invariance, In: Analytical Methods in Probability Theory (D. Dugué et al., eds.), Lecture Notes in Mathematics, 861, pp. 128–145. Springer, Berlin, 1981.
P. Révész, Random Walk in Random and Non-Random Environments. World Scientific, Singapore, 1990.
B. Tóth, Multiple covering of the range of a random walk on Z (on a question of P. Erdős and P. Révész), Studia Sci. Math. Hungar. 31 (1996), 355–359.
B. Tóth, No more than three favourite sites for simple random walk, Ann. Probab., to appear.
B. Tóth, and W. Werner, Tied favourite edges for simple random walk, Combin. Probab. Comput. 6 (1997), 359–369.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Shi, Z., Tóth, B. Favourite sites of simple random walk. Periodica Mathematica Hungarica 41, 237–249 (2000). https://doi.org/10.1023/A:1010389026544
Issue Date:
DOI: https://doi.org/10.1023/A:1010389026544