Electronic Communications in Probability

Slowdown estimates for one-dimensional random walks in random environment with holding times

Amir Dembo, Ryoki Fukushima, and Naoki Kubota

Full-text: Open access

Abstract

We consider a one dimensional random walk in random environment that is uniformly biased to one direction. In addition to the transition probability, the jump rate of the random walk is assumed to be spatially inhomogeneous and random. We study the probability that the random walk travels slower than its typical speed and determine its decay rate asymptotic.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 89, 12 pp.

Dates
Received: 3 July 2018
Accepted: 8 November 2018
First available in Project Euclid: 24 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1543028986

Digital Object Identifier
doi:10.1214/18-ECP191

Subjects
Primary: 60K37: Processes in random environments 60F10: Large deviations 60J15

Keywords
random walk random environment large deviation slowdown

Rights
Creative Commons Attribution 4.0 International License.

Citation

Dembo, Amir; Fukushima, Ryoki; Kubota, Naoki. Slowdown estimates for one-dimensional random walks in random environment with holding times. Electron. Commun. Probab. 23 (2018), paper no. 89, 12 pp. doi:10.1214/18-ECP191. https://projecteuclid.org/euclid.ecp/1543028986


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References

  • [1] S. W. Ahn and J. Peterson. Oscillations of quenched slowdown asymptotics for ballistic one-dimensional random walk in a random environment. Electron. J. Probab., 21(16):1–27, 2016.
  • [2] N. Berger. Slowdown estimates for ballistic random walk in random environment. J. Eur. Math. Soc. (JEMS), 14(1):127–174, 2012.
  • [3] F. Comets, N. Gantert, and O. Zeitouni. Quenched, annealed and functional large deviations for one-dimensional random walk in random environment. Probability Theory and Related Fields, 118(1):65–114, 2000.
  • [4] A. Dembo. Favorite points, cover times and fractals. In Lectures on probability theory and statistics, volume 1869 of Lecture Notes in Math., pages 1–101. Springer, Berlin, 2005.
  • [5] A. Dembo, N. Gantert, and O. Zeitouni. Large deviations for random walk in random environment with holding times. Ann. Probab., 32(1B):996–1029, 2004.
  • [6] A. Dembo, Y. Peres, and O. Zeitouni. Tail estimates for one-dimensional random walk in random environment. Comm. Math. Phys., 181(3):667–683, 1996.
  • [7] P. Embrechts, C. Klüppelberg, and T. Mikosch. Modelling extremal events, volume 33 of Applications of Mathematics (New York). Springer-Verlag, Berlin, 1997. For insurance and finance.
  • [8] N. Gantert and O. Zeitouni. Quenched sub-exponential tail estimates for one-dimensional random walk in random environment. Comm. Math. Phys., 194(1):177–190, 1998.
  • [9] N. Gantert and O. Zeitouni. Large deviations for one-dimensional random walk in a random environment—a survey. In Random walks (Budapest, 1998), volume 9 of Bolyai Soc. Math. Stud., pages 127–165. János Bolyai Math. Soc., Budapest, 1999.
  • [10] A. Greven and F. den Hollander. Large deviations for a random walk in random environment. Ann. Probab., 22(3):1381–1428, 1994.
  • [11] S. V. Nagaev. Integral limit theorems taking large deviations into account when Cramér’s condition does not hold. i. Theory Probab. Appl., 14(1):51–64, 1969.
  • [12] S. V. Nagaev. Integral limit theorems taking large deviations into account when Cramér’s condition does not hold. ii. Theory Probab. Appl., 14(2):193–208, 1969.
  • [13] S. V. Nagaev. Large deviations of sums of independent random variables. Ann. Probab., 7(5):745–789, 1979.
  • [14] A. Pisztora and T. Povel. Large deviation principle for random walk in a quenched random environment in the low speed regime. Ann. Probab., 27(3):1389–1413, 1999.
  • [15] A. Pisztora, T. Povel, and O. Zeitouni. Precise large deviation estimates for a one-dimensional random walk in a random environment. Probab. Theory Related Fields, 113(2):191–219, 1999.
  • [16] S. I. Resnick and R. J. Tomkins. Almost sure stability of maxima. J. Appl. Probability, 10:387–401, 1973.
  • [17] L. V. Rozovskiĭ. Probabilities of large deviations on the whole axis. Teor. Veroyatnost. i Primenen., 38(1):79–109, 1993.
  • [18] E. Seneta. Regularly varying functions. Lecture Notes in Mathematics, Vol. 508. Springer-Verlag, Berlin, 1976.
  • [19] F. Solomon. Random walks in a random environment. Ann. Probab., 3(1):1–31, 02 1975.
  • [20] A.-S. Sznitman. Slowdown and neutral pockets for a random walk in random environment. Probab. Theory Related Fields, 115(3):287–323, 1999.
  • [21] A.-S. Sznitman. Slowdown estimates and central limit theorem for random walks in random environment. J. Eur. Math. Soc. (JEMS), 2(2):93–143, 2000.
  • [22] R. J. Tomkins. Regular variation and the stability of maxima. Ann. Probab., 14(3):984–995, 1986.
  • [23] R. van der Hofstad, P. Mörters, and N. Sidorova. Weak and almost sure limits for the parabolic Anderson model with heavy tailed potentials. Ann. Appl. Probab., 18(6):2450–2494, 2008.