The Annals of Probability

Three favorite sites occurs infinitely often for one-dimensional simple random walk

Jian Ding and Jianfei Shen

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

For a one-dimensional simple random walk $(S_{t})$, for each time $t$ we say a site $x$ is a favorite site if it has the maximal local time. In this paper, we show that with probability 1 three favorite sites occurs infinitely often. Our work is inspired by Tóth [Ann. Probab. 29 (2001) 484–503], and disproves a conjecture of Erdős and Révész [In Mathematical Structure—Computational Mathematics—Mathematical Modelling 2 (1984) 152–157] and of Tóth [Ann. Probab. 29 (2001) 484–503].

Article information

Source
Ann. Probab., Volume 46, Number 5 (2018), 2545-2561.

Dates
Received: March 2017
Revised: September 2017
First available in Project Euclid: 24 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1535097634

Digital Object Identifier
doi:10.1214/17-AOP1232

Mathematical Reviews number (MathSciNet)
MR3846833

Zentralblatt MATH identifier
06964343

Subjects
Primary: 60J15 60J55: Local time and additive functionals

Keywords
Random walk favorite sites

Citation

Ding, Jian; Shen, Jianfei. Three favorite sites occurs infinitely often for one-dimensional simple random walk. Ann. Probab. 46 (2018), no. 5, 2545--2561. doi:10.1214/17-AOP1232. https://projecteuclid.org/euclid.aop/1535097634


Export citation

References

  • [1] Abe, Y. (2015). Maximum and minimum of local times for two-dimensional random walk. Electron. Commun. Probab. 20 22, 14 pp.
  • [2] Bass, R. F., Eisenbaum, N. and Shi, Z. (2000). The most visited sites of symmetric stable processes. Probab. Theory Related Fields 116 391–404.
  • [3] Bass, R. F. and Griffin, P. S. (1985). The most visited site of Brownian motion and simple random walk. Z. Wahrsch. Verw. Gebiete 70 417–436.
  • [4] Belius, D. (2013). Gumbel fluctuations for cover times in the discrete torus. Probab. Theory Related Fields 157 635–689.
  • [5] Belius, D. and Kistler, N. (2017). The subleading order of two dimensional cover times. Probab. Theory Related Fields 167 461–552.
  • [6] Chen, D., de Raphélis, L. and Hu, Y. Favorite sites of randomly biased walks on a supercritical Galton–Watson tree. Available at arXiv:1611.04497.
  • [7] Csáki, E., Révész, P. and Shi, Z. (2000). Favourite sites, favourite values and jump sizes for random walk and Brownian motion. Bernoulli 6 951–975.
  • [8] Csáki, E. and Shi, Z. (1998). Large favourite sites of simple random walk and the Wiener process. Electron. J. Probab. 3 14, 31 pp. (electronic).
  • [9] Dembo, A. (2005). Favorite points, cover times and fractals. In Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1869 1–101. Springer, Berlin.
  • [10] Dembo, A., Peres, Y., Rosen, J. and Zeitouni, O. (2001). Thick points for planar Brownian motion and the Erdős–Taylor conjecture on random walk. Acta Math. 186 239–270.
  • [11] Dembo, A., Peres, Y., Rosen, J. and Zeitouni, O. (2004). Cover times for Brownian motion and random walks in two dimensions. Ann. of Math. (2) 160 433–464.
  • [12] Eisenbaum, N. (1997). On the most visited sites by a symmetric stable process. Probab. Theory Related Fields 107 527–535.
  • [13] Eisenbaum, N. and Khoshnevisan, D. (2002). On the most visited sites of symmetric Markov processes. Stochastic Process. Appl. 101 241–256.
  • [14] Erdős, P. and Révész, P. (1987). Problems and results on random walks. In Mathematical Statistics and Probability Theory, Vol. B (Bad Tatzmannsdorf, 1986) 59–65. Reidel, Dordrecht.
  • [15] Erdős, P. and Révész, P. (1984). On the favourite points of a random walk. In Mathematical Structure—Computational Mathematics—Mathematical Modelling 2 152–157.
  • [16] Erdős, P. and Révész, P. (1991). Three problems on the random walk in $\mathbf{Z}^{d}$. Studia Sci. Math. Hungar. 26 309–320.
  • [17] Hu, Y. and Shi, Z. (2000). The problem of the most visited site in random environment. Probab. Theory Related Fields 116 273–302.
  • [18] Hu, Y. and Shi, Z. (2015). The most visited sites of biased random walks on trees. Electron. J. Probab. 20 62, 14 pp.
  • [19] Knight, F. B. (1963). Random walks and a sojourn density process of Brownian motion. Trans. Amer. Math. Soc. 109 56–86.
  • [20] Lifshits, M. A. and Shi, Z. (2004). The escape rate of favorite sites of simple random walk and Brownian motion. Ann. Probab. 32 129–152.
  • [21] Marcus, M. B. (2001). The most visited sites of certain Lévy processes. J. Theoret. Probab. 14 867–885.
  • [22] Okada, I. Topics and problems on favorite sites of random walks. Available at arXiv:1606.03787.
  • [23] Shi, Z. and Tóth, B. (2000). Favourite sites of simple random walk. Period. Math. Hungar. 41 237–249. Endre Csáki 65.
  • [24] Tóth, B. (2001). No more than three favorite sites for simple random walk. Ann. Probab. 29 484–503.
  • [25] Tóth, B. and Werner, W. (1997). Tied favourite edges for simple random walk. Combin. Probab. Comput. 6 359–369.