Electronic Communications in Probability

A functional limit theorem for excited random walks

Andrey Pilipenko

Full-text: Open access

Abstract

We consider the limit behavior of an excited random walk (ERW), i.e., a random walk whose transition probabilities depend on the number of times the walk has visited to the current state. We prove that an ERW being naturally scaled converges in distribution to an excited Brownian motion that satisfies an SDE, where the drift of the unknown process depends on its local time. Similar result was obtained by Raimond and Schapira, their proof was based on the Ray-Knight type theorems. We propose a new method based on a study of the Radon-Nikodym density of the ERW distribution with respect to the distribution of a symmetric random walk.

Article information

Source
Electron. Commun. Probab., Volume 22 (2017), paper no. 39, 9 pp.

Dates
Received: 11 November 2016
Accepted: 14 June 2017
First available in Project Euclid: 9 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1502244193

Digital Object Identifier
doi:10.1214/17-ECP66

Mathematical Reviews number (MathSciNet)
MR3685237

Zentralblatt MATH identifier
06797792

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments 60F17: Functional limit theorems; invariance principles

Keywords
excited random walks excited Brownian motion invariance principle

Rights
Creative Commons Attribution 4.0 International License.

Citation

Pilipenko, Andrey. A functional limit theorem for excited random walks. Electron. Commun. Probab. 22 (2017), paper no. 39, 9 pp. doi:10.1214/17-ECP66. https://projecteuclid.org/euclid.ecp/1502244193


Export citation

References

  • [1] Basdevant, A. L. and Singh, A.: On the speed of a cookie random walk. Probability Theory and Related Fields 141(3–4), (2008), 625–645.
  • [2] Billingsley, P.: Convergence of probability measures. Wiley John Wiley & Sons, Inc., New York-London-Sydney, 1968. xii+253 pp.
  • [3] Borodin, A. N.: An asymptotic behaviour of local times of a recurrent random walk with finite variance. Theory Probab. Appl. 26(4), (1982), 758–772.
  • [4] Chaumont, L. and Doney, R. A.: Pathwise uniqueness for perturbed versions of Brownian motion and reflected Brownian motion. Probability theory and related fields 113(4), (1999), 519–534.
  • [5] Dolgopyat, D.: Central limit theorem for excited random walk in the recurrent regime. ALEA Lat. Am. J. Probab. Math. Stat. 8, (2011), 259–268.
  • [6] Dolgopyat, D. and Kosygina, E.: Scaling limits of recurrent excited random walks on integers. Electron. Commun. Probab. 17(35), (2012), 1–14.
  • [7] Gikhman, I. I. and Skorokhod, A. V.: On the densities of probability measures in function spaces. Russian Mathematical Surveys 21(6), (1966), 83–156.
  • [8] Kosygina, E. and Zerner, M. P.: Positively and negatively excited random walks on integers, with branching processes. Electron. J. Probab. 13(64), (2008), 1952–1979.
  • [9] Kosygina, E. and Zerner, M. P.: Excited random walks: results, methods, open problems. Excited random walks: results, methods, open problems. Bull. Inst. Math. Acad. Sin. (N.S.). 8(1), (2013), 105–157 (in a special issue in honor of S.R.S. Varadhan’s 70th birthday).
  • [10] Liptser, R., and Shiryaev, A. N.: Statistics of random processes. (Russian) Nauka, Moscow, 1974. 696 pp.
  • [11] Merkl, F. and Rolles, S. W.: Linearly edge-reinforced random walks. Lecture Notes – Monograph Series, (2006), 66–77.
  • [12] Norris, J. R., Rogers, L. C. G., and Williams, D.: Self-avoiding random walk: a Brownian motion model with local time drift. Probability Theory and Related Fields 74(2), (1987), 271–287.
  • [13] Pemantle, R. and Volkov, S.: Vertex-reinforced random walk on Z has finite range. The Annals of Probability 27(3), (1999), 1368–1388.
  • [14] Pilipenko, A. and Khomenko, V.: On a limit behavior of a random walk with modifications at zero. arXiv:1611.02048
  • [15] Raimond, O. and Schapira, B.: Excited Brownian motions as limits of excited random walks. Probability Theory and Related Fields 154(3–4), (2012), 875–909.
  • [16] Raimond, O. and Schapira, B.: Excited Brownian motions. ALEA Lat. Am. J. Probab. Math. Stat. 8, (2011), 19–41.
  • [17] Skorokhod, A. V. Studies in the theory of random processes. Addison-Wesley Publishing Company, 1965. viii+199 pp.
  • [18] Zerner, M. P.: Multi-excited random walks on integers. Probability theory and related fields 133(1), (2005), 98–122.