Weak quenched limiting distributions for transient one-dimensional random walk in a random environment

Jonathon Peterson; Gennady Samorodnitsky

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 3, page 722-752
  • ISSN: 0246-0203

Abstract

top
We consider a one-dimensional, transient random walk in a random i.i.d. environment. The asymptotic behaviour of such random walk depends to a large extent on a crucial parameter κ g t ; 0 that determines the fluctuations of the process. When 0 l t ; κ l t ; 2 , the averaged distributions of the hitting times of the random walk converge to a κ -stable distribution. However, it was shown recently that in this case there does not exist a quenched limiting distribution of the hitting times. That is, it is not true that for almost every fixed environment, the distributions of the hitting times (centered and scaled in any manner) converge to a non-degenerate distribution. We show, however, that the quenched distributions do have a limit in the weak sense. That is, the quenched distributions of the hitting times – viewed as a random probability measure on ℝ – converge in distribution to a random probability measure, which has interesting stability properties. Our results generalize both the averaged limiting distribution and the non-existence of quenched limiting distributions.

How to cite

top

Peterson, Jonathon, and Samorodnitsky, Gennady. "Weak quenched limiting distributions for transient one-dimensional random walk in a random environment." Annales de l'I.H.P. Probabilités et statistiques 49.3 (2013): 722-752. <http://eudml.org/doc/272096>.

@article{Peterson2013,
abstract = {We consider a one-dimensional, transient random walk in a random i.i.d. environment. The asymptotic behaviour of such random walk depends to a large extent on a crucial parameter $\kappa &gt;0$ that determines the fluctuations of the process. When $0&lt;\kappa &lt;2$, the averaged distributions of the hitting times of the random walk converge to a $\kappa $-stable distribution. However, it was shown recently that in this case there does not exist a quenched limiting distribution of the hitting times. That is, it is not true that for almost every fixed environment, the distributions of the hitting times (centered and scaled in any manner) converge to a non-degenerate distribution. We show, however, that the quenched distributions do have a limit in the weak sense. That is, the quenched distributions of the hitting times – viewed as a random probability measure on $\mathbb \{R\}$ – converge in distribution to a random probability measure, which has interesting stability properties. Our results generalize both the averaged limiting distribution and the non-existence of quenched limiting distributions.},
author = {Peterson, Jonathon, Samorodnitsky, Gennady},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {weak quenched limits; point processes; Heavy tails; heavy tails},
language = {eng},
number = {3},
pages = {722-752},
publisher = {Gauthier-Villars},
title = {Weak quenched limiting distributions for transient one-dimensional random walk in a random environment},
url = {http://eudml.org/doc/272096},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Peterson, Jonathon
AU - Samorodnitsky, Gennady
TI - Weak quenched limiting distributions for transient one-dimensional random walk in a random environment
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 3
SP - 722
EP - 752
AB - We consider a one-dimensional, transient random walk in a random i.i.d. environment. The asymptotic behaviour of such random walk depends to a large extent on a crucial parameter $\kappa &gt;0$ that determines the fluctuations of the process. When $0&lt;\kappa &lt;2$, the averaged distributions of the hitting times of the random walk converge to a $\kappa $-stable distribution. However, it was shown recently that in this case there does not exist a quenched limiting distribution of the hitting times. That is, it is not true that for almost every fixed environment, the distributions of the hitting times (centered and scaled in any manner) converge to a non-degenerate distribution. We show, however, that the quenched distributions do have a limit in the weak sense. That is, the quenched distributions of the hitting times – viewed as a random probability measure on $\mathbb {R}$ – converge in distribution to a random probability measure, which has interesting stability properties. Our results generalize both the averaged limiting distribution and the non-existence of quenched limiting distributions.
LA - eng
KW - weak quenched limits; point processes; Heavy tails; heavy tails
UR - http://eudml.org/doc/272096
ER -

References

top
  1. [1] S. Alili. Asymptotic behaviour for random walks in random environments. J. Appl. Probab.36 (1999) 334–349. Zbl0946.60046MR1724844
  2. [2] P. Billingsley. Convergence of Probability Measures, 2nd edition. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York, 1999. Zbl0172.21201MR1700749
  3. [3] L. Breiman. On some limit theorems similar to the arc-sine law. Theory Probab. Appl.10 (1965) 323–331. Zbl0147.37004MR184274
  4. [4] R. A. Davis and T. Hsing. Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Probab.23 (1995) 879–917. Zbl0837.60017MR1334176
  5. [5] D. Dolgopyat and I. Goldsheid. Quenched limit theorems for nearest neighbour random walks in 1d random environment. Preprint, 2010. Available at arXiv:1012.2503v1. Zbl1260.60187MR2966946
  6. [6] N. Enriquez, C. Sabot and O. Zindy. Aging and quenched localization for one-dimensional random walks in random environment in the sub-ballistic regime. Bull. Soc. Math. France137 (2009) 423–452. Zbl1186.60108MR2574090
  7. [7] N. Enriquez, C. Sabot and O. Zindy. Limit laws for transient random walks in random environment on ℤ . Ann. Inst. Fourier (Grenoble) 59 (2009) 2469–2508. Zbl1200.60093MR2640927
  8. [8] N. Enriquez, C. Sabot, L. Tournier and O. Zindy. Annealed and quenched fluctuations for ballistic random walks in random environment on ℤ . Preprint, 2010. Available at arXiv:1012.1959v1. Zbl1279.60126
  9. [9] N. Gantert and Z. Shi. Many visits to a single site by a transient random walk in random environment. Stochastic Process. Appl.99 (2002) 159–176. Zbl1059.60100MR1901151
  10. [10] I. Ya. Goldsheid. Simple transient random walks in one-dimensional random environment: The central limit theorem. Probab. Theory Related Fields139 (2007) 41–64. Zbl1134.60065MR2322691
  11. [11] H. Kesten, M. V. Kozlov and F. Spitzer. A limit law for random walk in a random environment. Compos. Math.30 (1975) 145–168. Zbl0388.60069MR380998
  12. [12] J. Peterson. Limiting distributions and large deviations for random walks in random environments. Ph.D. thesis, Univ. Minnesota, 2008. Available at arXiv:0810.0257v1. MR2711962
  13. [13] J. Peterson. Quenched limits for transient, ballistic, sub-Gaussian one-dimensional random walk in random environment. Ann. Inst. Henri Poincaré Probab. Stat.45 (2009) 685–709. Zbl1178.60071MR2548499
  14. [14] J. Peterson and O. Zeitouni. Quenched limits for transient, zero speed one-dimensional random walk in random environment. Ann. Probab.37 (2009) 143–188. Zbl1179.60070MR2489162
  15. [15] S. I. Resnick. Extreme Values, Regular Variation and Point Processes. Springer Series in Operations Research and Financial Engineering. Springer, New York, 2008. Zbl1136.60004MR2364939
  16. [16] G. Samorodnitsky and M. S. Taqqu. Stable Non-Gaussian Random Processes. Chapman & Hall, New York, 1994. Zbl0925.60027
  17. [17] T. Shiga and H. Tanaka. Infinitely divisible random probability distributions with an application to a random motion in a random environment. Electron. J. Probab. 11 (2006) 1144–1183 (electronic). Zbl1127.60051MR2268541
  18. [18] F. Solomon. Random walks in a random environment. Ann. Probability3 (1975) 1–31. Zbl0305.60029MR362503
  19. [19] O. Zeitouni. Random walks in random environment. In Lectures on Probability Theory and Statistics 189–312. Lecture Notes in Math. 1837. Springer, Berlin, 2004. Zbl1060.60103MR2071631

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.