Skip to main content

Limit Theorems for the One-Dimensional Random Walk with Random Resetting to the Maximum


The first part of this paper is devoted to study a model of one-dimensional random walk with memory to the maximum position described as follows. At each step the walker resets to the rightmost visited site with probability \(r \in (0,1)\) and moves as the simple random walk with remaining probability. Using the approach of renewal theory, we prove the laws of large numbers and the central limit theorems for the random walk. These results reprove and significantly enhance the analysis of the mean value and variance of the process established in Majumdar et al. (Phys Rev E 92:052126, 2015). In the second part, we expand the analysis to the situation where the memory of the walker decreases over time by assuming that at the step n the resetting probability is \(r_n = \min \{rn^{-a}, \tfrac{1}{2}\}\) with ra positive parameters. For this model, we first establish the asymptotic behavior of the mean values of \(X_n\)-the current position and \(M_n\)-the maximum position of the random walk. As a consequence, we observe an interesting phase transition of the ratio \({{\mathbb {E}}}[X_n]/{{\mathbb {E}}}[M_n]\) when a varies. Precisely, it converges to 1 in the subcritical phase \(a\in (0,1)\), to a constant \(c\in (0,1)\) in the critical phase \(a=1\), and to 0 in the supercritical phase \(a>1\). Finally, when \(a>1\), we show that the model behaves closely to the simple random walk in the sense that \(\frac{X_n}{\sqrt{n}} \overset{(d)}{\longrightarrow } {\mathcal {N}}(0,1)\) and \(\frac{M_n}{\sqrt{n}} \overset{(d)}{\longrightarrow } \max _{0 \le t \le 1} B_t\), where \({\mathcal {N}}(0,1)\) is the standard normal distribution and \((B_t)_{t\ge 0}\) is the standard Brownian motion.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4


  1. 1.

    Bartumeus, F., Catalan, J.: Optimal search behaviour and classic foraging theory. J. Phys. A: Math. Theor. 42, 434002 (2009)

    ADS  Article  Google Scholar 

  2. 2.

    Bartumeus, F., da Luz, M.G.E., Viswanathan, G.M., Catalan, J.: Animal search strategies: a quantitative random-walk analysis. Ecology 86(11), 3078–3087 (2005)

    Article  Google Scholar 

  3. 3.

    Bénichou, O., Loverdo, C., Moreau, M., Voituriez, R.: Intermittent search strategies. Rev. Mod. Phys. 83, 81 (2011)

    ADS  Article  Google Scholar 

  4. 4.

    Boyer, D., Solis-Salas, C.: Random walks with preferential relocation to places visited in the past and their application to biology. Phys. Rev. Lett. 112, 240601 (2014)

    ADS  Article  Google Scholar 

  5. 5.

    Evans, M.R., Majumdar, S.N.: Diffusion with stochastic resetting. Phys. Rev. Lett. 106, 160601 (2011)

    ADS  Article  Google Scholar 

  6. 6.

    Evans, M.R., Majumdar, S.N., Schehr, G.: Stochastic resetting and applications. J. Phys. A: Math. Theor 53, 193001 (2020)

    ADS  MathSciNet  Article  Google Scholar 

  7. 7.

    Gut, A.: Stopped Random Walks: Limit Theorem and Applications. Springer, New York (2009)

    Book  Google Scholar 

  8. 8.

    Gautestad, A.O., Mysterud, I.: Intrinsic scaling complexity in animal dispersion and abundance. Am. Nat. 165, 44 (2005)

    Article  Google Scholar 

  9. 9.

    Giuggioli, L., Gupta, S., Chase, M.: Comparison of two models of tethered motion. J. Phys. A: Math. Theor 52(7), 075001 (2019)

    ADS  Article  Google Scholar 

  10. 10.

    Kusmierz, L., Majumdar, S.N., Sabhapandit, S., Schehr, G.: First order transition for the optimal search time of Lévy flights with resetting. Phys. Rev. Lett. 113, 220602 (2014)

    ADS  Article  Google Scholar 

  11. 11.

    Lawler, G.F., Limic, V.: Random Walk: A Modern Introduction. Cambridge Press, Cambridge (2010)

    Book  Google Scholar 

  12. 12.

    Luby, M., Sinclair, A., Zuckerman, D.: Optimal speedup of Las Vegas algorithms. Inf. Process. Lett. 47(4), 173–180 (1993)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Majumdar, S.N., Sabhapandit, S., Schehr, G.: Random walk with random resetting to the maximum position. Phys. Rev. E 92, 052126 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  14. 14.

    Montanari, A., Zecchina, R.: Optimizing searches via rare events. Phys. Rev. Lett. 88, 178701 (2002)

    ADS  Article  Google Scholar 

  15. 15.

    Montero, M., Villarroel, J.: Monotonic continuous-time random walks with drift and stochastic reset events. Phys. Rev. E 87, 012116 (2013)

    ADS  Article  Google Scholar 

  16. 16.

    Reuveni, S., Urbakh, M., Klafter, J.: Role of substrate unbinding in Michaelis–Menten enzymatic reactions. Proc. Natl. Acad. Sci. USA 111, 4391 (2014)

    ADS  Article  Google Scholar 

  17. 17.

    Roldán, E., Lisica, A., Sánchez-Taltavull, D., Grill, S.W.: Stochastic resetting in backtrack recovery by RNA polymerases. Phys. Rev. E 93(6), 062411 (2016)

    ADS  Article  Google Scholar 

  18. 18.

    Rotbart, T., Reuveni, S., Urbakh, M.: Michaelis–Menten reaction scheme as a unified approach towards the optimal restart problem. Phys. Rev. E 92, 060101(R) (2015)

    ADS  Article  Google Scholar 

  19. 19.

    Viswanathan, G.M., da Luz, M.G.E., Raposo, E.P., Stanley, H.: The Physics of Foraging: An Introduction to Random Searches and Biological Encounters. Cambridge University Press, Cambridge (2011)

    Book  Google Scholar 

Download references


The work of Van Hao Can is supported by the Singapore Ministry of Education Academic Research Fund Tier 2 Grant MOE2018-T2-2-076. Van Quyet Nguyen is supported by International Center for Research and Postgraduate Training in Mathematics, Institute of Mathematics, Vietnam Academy of Science and Technology under grant number ICRTM03-2020.02. The work of Thai Son Doan is funded by Vietnam Ministry of Education and Training under Grant Number B2021-TDV-01.

Author information



Corresponding author

Correspondence to Van Hao Can.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Satya Majumdar.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Can, V.H., Doan, T.S. & Nguyen, V.Q. Limit Theorems for the One-Dimensional Random Walk with Random Resetting to the Maximum. J Stat Phys 183, 21 (2021).

Download citation


  • Limit theorems
  • Random walk
  • Stochastic resetting

Mathematics Subject Classification

  • Primary 60G50
  • Secondary 60J10