Quenched limits for transient, ballistic, sub-gaussian one-dimensional random walk in random environment
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 3, pp. 685-709.

On examine des marches aléatoires unidimensionnelles en milieu aléatoire avec un environnement i.i.d., dans le régime où la marche est transiente avec vitesse vP>0 et où il existe s∈(1, 2) tel que la loi «annealed» (i.e., moyennée) de n-1/s(Xn-nvP) converge vers une loi stable de paramètre s. Sous la loi «quenched» (i.e. conditionnelement à l'environnement) on montre qu'il n'existe pas de loi limite. En particulier on prouve qu'il existe des suites {tk} et {tk'}, dépendant de l'environnement, tel qu'un théorème de limite centrale quenched est valide le long de la suite tk, mais où la distribution limite suivant la suite tk' est une distribution centrée exponentielle inverse. Ceci complète les résultats d'un article récent de Peterson et Zeitouni (arXiv:math/0704.1778v1 [math.PR]) qui traitait le case de paramètre s∈(0, 1).

We consider a nearest-neighbor, one-dimensional random walk {Xn}n≥0 in a random i.i.d. environment, in the regime where the walk is transient with speed vP>0 and there exists an s∈(1, 2) such that the annealed law of n-1/s(Xn-nvP) converges to a stable law of parameter s. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible. In particular we show that there exist sequences {tk} and {tk'} depending on the environment only, such that a quenched central limit theorem holds along the subsequence tk, but the quenched limiting distribution along the subsequence tk' is a centered reverse exponential distribution. This complements the results of a recent paper of Peterson and Zeitouni (arXiv:math/0704.1778v1 [math.PR]) which handled the case when the parameter s∈(0, 1).

DOI : 10.1214/08-AIHP149
Classification : 60K37, 60F05, 82C41, 82D30
Mots clés : random walk, random environment
@article{AIHPB_2009__45_3_685_0,
     author = {Peterson, Jonathon},
     title = {Quenched limits for transient, ballistic, sub-gaussian one-dimensional random walk in random environment},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {685--709},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {3},
     year = {2009},
     doi = {10.1214/08-AIHP149},
     mrnumber = {2548499},
     zbl = {1178.60071},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/08-AIHP149/}
}
TY  - JOUR
AU  - Peterson, Jonathon
TI  - Quenched limits for transient, ballistic, sub-gaussian one-dimensional random walk in random environment
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2009
SP  - 685
EP  - 709
VL  - 45
IS  - 3
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/08-AIHP149/
DO  - 10.1214/08-AIHP149
LA  - en
ID  - AIHPB_2009__45_3_685_0
ER  - 
%0 Journal Article
%A Peterson, Jonathon
%T Quenched limits for transient, ballistic, sub-gaussian one-dimensional random walk in random environment
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2009
%P 685-709
%V 45
%N 3
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/08-AIHP149/
%R 10.1214/08-AIHP149
%G en
%F AIHPB_2009__45_3_685_0
Peterson, Jonathon. Quenched limits for transient, ballistic, sub-gaussian one-dimensional random walk in random environment. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 3, pp. 685-709. doi : 10.1214/08-AIHP149. http://archive.numdam.org/articles/10.1214/08-AIHP149/

[1] Y. S. Chow and H. Teicher. Probability Theory: Independence, Interchangeability, Martingales. Springer, New York, 1978. | MR | Zbl

[2] N. Enriquez, C. Sabot and O. Zindy. Limit laws for transient random walks in random environment on ℤ. Preprint, 2007. Available at arXiv:math/0703660v3 [math.PR].

[3] I. Y. Goldsheid. Simple transient random walks in one-dimensional random environment: The central limit theorem, 2006. Available at math.PR/0605775 [math.PR]. | MR

[4] D. L. Iglehart. Extreme values in the GI/G/1 queue. Ann. Math. Statist. 43 (1972) 627-635. | MR | Zbl

[5] 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

[6] M. Kobus. Generalized Poisson distributions as limits of sums for arrays of dependent random vectors. J. Multivariate Anal. 52 (1995) 199-244. | MR | Zbl

[7] S. M. Kozlov and S. A. Molchanov. Conditions for the applicability of the central limit theorem to random walks on a lattice (Russian). Dokl. Akad. Nauk SSSR 278 (1984) 531-534. | MR | Zbl

[8] J. Peterson. Limiting distributions and large devations for random walks in random enviroments. Ph.D. thesis, University of Minnesota, 2008. Available at arXiv:math0810.0257v1.

[9] 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

[10] F. Solomon. Random walks in random environments. Ann. Probab. 3 (1975) 1-31. | MR | Zbl

[11] 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

Cité par Sources :