Electronic Journal of Probability

On some generalized reinforced random walk on integers

Olivier Raimond and Bruno Schapira

Full-text: Open access


We consider Reinforced Random Walks where transitions probabilities are a function of the proportions of times the walk has traversed an edge. We give conditions for recurrence or transience. A phase transition is observed, similar to Pemantle [7] on trees

Article information

Electron. J. Probab., Volume 14 (2009), paper no. 60, 1770-1789.

Accepted: 24 August 2009
First available in Project Euclid: 1 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F20: Zero-one laws

Reinforced random walks urn processes

This work is licensed under aCreative Commons Attribution 3.0 License.


Raimond, Olivier; Schapira, Bruno. On some generalized reinforced random walk on integers. Electron. J. Probab. 14 (2009), paper no. 60, 1770--1789. doi:10.1214/EJP.v14-685. https://projecteuclid.org/euclid.ejp/1464819521

Export citation


  • Benaïm, Michel. Dynamics of stochastic approximation algorithms. Séminaire de Probabilités, XXXIII, 1–68, Lecture Notes in Math., 1709, Springer, Berlin, 1999.
  • Benaim, Michel; Hirsch, Morris W. Dynamics of Morse-Smale urn processes. Ergodic Theory Dynam. Systems 15 (1995), no. 6, 1005–1030.
  • Coppersmith D., Diaconis P.: Random walk with reinforcement, unpublished, (1987).
  • Duflo, Marie. Algorithmes stochastiques.(French) [Stochastic algorithms] Mathématiques & Applications (Berlin) [Mathematics & Applications], 23. Springer-Verlag, Berlin, 1996. xiv+319 pp. ISBN: 3-540-60699-8
  • Hill B., Lane D., and Sudderth W.: A strong law for some generalized urn processes, Ann. Probab. 8, (1980), 214–226.
  • Kosygina, Elena; Zerner, Martin P. W. Positively and negatively excited random walks on integers, with branching processes. Electron. J. Probab. 13 (2008), no. 64, 1952–1979.
  • Pemantle, Robin. Phase transition in reinforced random walk and RWRE on trees. Ann. Probab. 16 (1988), no. 3, 1229–1241.
  • Pemantle R.: Random processes with reinforcement, Doctoral dissertation, M.I.T., (1988).
  • Pemantle, Robin. Nonconvergence to unstable points in urn models and stochastic approximations. Ann. Probab. 18 (1990), no. 2, 698–712.
  • Pemantle, Robin. When are touchpoints limits for generalized Pólya urns? Proc. Amer. Math. Soc. 113 (1991), no. 1, 235–243.
  • Pemantle, Robin. A survey of random processes with reinforcement. Probab. Surv. 4 (2007), 1–79 (electronic).
  • Williams, David. Probability with martingales.Cambridge Mathematical Textbooks. Cambridge University Press, Cambridge, 1991. xvi+251 pp. ISBN: 0-521-40455-X; 0-521-40605-6
  • Zerner, Martin P. W. Multi-excited random walks on integers. Probab. Theory Related Fields 133 (2005), no. 1, 98–122.
  • Zerner, Martin P. W. Recurrence and transience of excited random walks on $\Bbb Z\sp d$ and strips. Electron. Comm. Probab. 11 (2006), 118–128 (electronic).