Electronic Journal of Probability

Large deviations and slowdown asymptotics for one-dimensional excited random walks

Jonathon Peterson

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We study the large deviations of excited random walks on $\mathbb{Z}$. We prove a large deviation principle for both the hitting times and the position of the random walk and give a qualitative description of the respective rate functions. When the excited random walk is transient with positive speed $v_0$, then the large deviation rate function for the position of the excited random walk is zero on the interval $[0,v_0]$ and so probabilities such as $P(X_n < nv)$ for $v \in (0,v_0)$ decay subexponentially. We show that rate of decay for such slowdown probabilities is polynomial of the order $n^{1-\delta/2}$, where $\delta>2$ is the expected total drift per site of the cookie environment. 

Article information

Electron. J. Probab., Volume 17 (2012), paper no. 48, 24 pp.

Accepted: 21 June 2012
First available in Project Euclid: 4 June 2016

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60F10: Large deviations 60K37: Processes in random environments

excited random walk large deviations

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Peterson, Jonathon. Large deviations and slowdown asymptotics for one-dimensional excited random walks. Electron. J. Probab. 17 (2012), paper no. 48, 24 pp. doi:10.1214/EJP.v17-1726. https://projecteuclid.org/euclid.ejp/1465062370

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  • Basdevant, Anne-Laure; Singh, Arvind. On the speed of a cookie random walk. Probab. Theory Related Fields 141 (2008), no. 3-4, 625–645.
  • Basdevant, Anne-Laure; Singh, Arvind. Rate of growth of a transient cookie random walk. Electron. J. Probab. 13 (2008), no. 26, 811–851.
  • Benjamini, Itai; Wilson, David B. Excited random walk. Electron. Comm. Probab. 8 (2003), 86–92 (electronic).
  • Brillinger, David R. A note on the rate of convergence of a mean. Biometrika 49 1962 574–576.
  • Bryc, Włodzimierz; Dembo, Amir. Large deviations and strong mixing. Ann. Inst. H. Poincaré Probab. Statist. 32 (1996), no. 4, 549–569.
  • Comets, Francis; Gantert, Nina; Zeitouni, Ofer. Quenched, annealed and functional large deviations for one-dimensional random walk in random environment. Probab. Theory Related Fields 118 (2000), no. 1, 65–114.
  • de Acosta, A. Upper bounds for large deviations of dependent random vectors. Z. Wahrsch. Verw. Gebiete 69 (1985), no. 4, 551–565.
  • Dembo, Amir; Peres, Yuval; Zeitouni, Ofer. Tail estimates for one-dimensional random walk in random environment. Comm. Math. Phys. 181 (1996), no. 3, 667–683.
  • Dembo, Amir; Zeitouni, Ofer. Large deviations techniques and applications. Corrected reprint of the second (1998) edition. Stochastic Modelling and Applied Probability, 38. Springer-Verlag, Berlin, 2010. xvi+396 pp. ISBN: 978-3-642-03310-0
  • Dolgopyat, Dmitry. Central limit theorem for excited random walk in the recurrent regime. ALEA Lat. Am. J. Probab. Math. Stat. 8 (2011), 259–268.
  • Dmitry Dolgopyat and Elena Kosygina, Scaling limits of recurrent excited random walks on integers, January 2012, available at arXiv:math/1201.0379.
  • Kesten, H.; Kozlov, M. V.; Spitzer, F. A limit law for random walk in a random environment. Compositio Math. 30 (1975), 145–168.
  • Kosygina, Elena; Mountford, Thomas. Limit laws of transient excited random walks on integers. Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011), no. 2, 575–600.
  • 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.
  • Lawler, Gregory F. Introduction to stochastic processes. Second edition. Chapman & Hall/CRC, Boca Raton, FL, 2006. xiv+234 pp. ISBN: 978-1-58488-651-8; 1-58488-651-X
  • Mogul'skiĭ, A. A. Small deviations in the space of trajectories. (Russian) Teor. Verojatnost. i Primenen. 19 (1974), 755–765.
  • Nagaev, S. V. Large deviations of sums of independent random variables. Ann. Probab. 7 (1979), no. 5, 745–789.
  • Ney, P.; Nummelin, E. Markov additive processes II. Large deviations. Ann. Probab. 15 (1987), no. 2, 593–609.
  • Rassoul-Agha, Firas. Large deviations for random walks in a mixing random environment and other (non-Markov) random walks. Comm. Pure Appl. Math. 57 (2004), no. 9, 1178–1196.
  • Solomon, Fred. Random walks in a random environment. Ann. Probability 3 (1975), 1–31.
  • Zerner, Martin P. W. Multi-excited random walks on integers. Probab. Theory Related Fields 133 (2005), no. 1, 98–122.