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

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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

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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 -

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