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

Peterson, Jonathon; Samorodnitsky, Gennady

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 3, pp. 722-752.

Nous considérons une marche aléatoire unidimensionnelle dans un environnement i.i.d. Le comportement asymptotique d’une telle marche aléatoire dépend largement d’un paramètre crucial $\kappa$ qui détermine les fluctuations du processus. Si $0<\kappa <2$, alors les distributions moyennées des temps d’atteinte de la marche aléatoire convergent vers une loi $\kappa$-stable. Cependant, il a été récemment prouvé que dans ce cas là, il n’existe pas de distribution limite des temps d’atteinte à environnement fixé. C’est-à-dire, il n’est pas vrai que presque tout environnement fixé, les distributions des temps d’atteinte (centrés et normalisés de quelque manière que ce soit) convergent vers une distribution non dégénérée. Nous montrons néanmoins que les distributions à environnement fixé ont une limite au sens faible. Plus précisément, les distributions à environnement fixé des temps d’atteinte - vues comme des mesures de probabilité aléatoires sur $ℝ$ - convergent en distribution vers une mesure de probabilité aléatoire qui a d’intéressantes propriétés de stabilité. Nos résultats généralisent à la fois la limite des distributions moyennisées et la non existence de distributions limites à environnement fixé.

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 $\kappa >0$ that determines the fluctuations of the process. When $0<\kappa <2$, 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 $ℝ$ - 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.

Classification : 60K37, 60F05, 60G55
Mots clés : weak quenched limits, point processes, Heavy tails

[1] S. Alili. Asymptotic behaviour for random walks in random environments. J. Appl. Probab. 36 (1999) 334-349. | MR | Zbl

[2] P. Billingsley. Convergence of Probability Measures, 2nd edition. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York, 1999. | MR | Zbl

[3] L. Breiman. On some limit theorems similar to the arc-sine law. Theory Probab. Appl. 10 (1965) 323-331. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. France 137 (2009) 423-452. | Numdam | MR | Zbl

[7] N. Enriquez, C. Sabot and O. Zindy. Limit laws for transient random walks in random environment on $\mathbb{Z}$. Ann. Inst. Fourier (Grenoble) 59 (2009) 2469-2508. | Numdam | MR | Zbl

[8] N. Enriquez, C. Sabot, L. Tournier and O. Zindy. Annealed and quenched fluctuations for ballistic random walks in random environment on $\mathbb{Z}$. Preprint, 2010. Available at arXiv:1012.1959v1.

[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. | MR | Zbl

[10] I. Ya. Goldsheid. Simple transient random walks in one-dimensional random environment: The central limit theorem. Probab. Theory Related Fields 139 (2007) 41-64. | MR | Zbl

[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. | Numdam | MR | Zbl

[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. | MR

[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. | Numdam | MR | Zbl

[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. | MR | Zbl

[15] S. I. Resnick. Extreme Values, Regular Variation and Point Processes. Springer Series in Operations Research and Financial Engineering. Springer, New York, 2008. | MR | Zbl

[16] G. Samorodnitsky and M. S. Taqqu. Stable Non-Gaussian Random Processes. Chapman & Hall, New York, 1994. | Zbl

[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). | MR | Zbl

[18] F. Solomon. Random walks in a random environment. Ann. Probability 3 (1975) 1-31. | MR | Zbl

[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. | MR | Zbl