Almost sure functional central limit theorem for ballistic random walk in random environment
Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 2, p. 373-420

We consider a multidimensional random walk in a product random environment with bounded steps, transience in some spatial direction, and high enough moments on the regeneration time. We prove an invariance principle, or functional central limit theorem, under almost every environment for the diffusively scaled centered walk. The main point behind the invariance principle is that the quenched mean of the walk behaves subdiffusively.

Nous considérons une marche aléatoire multidimensionnelle en environnement aléatoire produit. La marche est à pas bornés, transiente dans une direction spatiale donnée, et telle que le temps de régénération posséde un moment suffisamment haut. Nous prouvons un principe d'invariance, ou un théorème limite central fonctionnel, sous presque tout environnement pour la marche centrée et diffusivement normalisée. Le point principal derrière le principe d'invariance est que la moyenne trempée (quenched) de la marche est sous-diffusive.

DOI : https://doi.org/10.1214/08-AIHP167
Classification:  60K37,  60F05,  60F17,  82D30
Keywords: random walk, ballistic, random environment, central limit theorem, invariance principle, point of view of the particle, environment process, Green function
@article{AIHPB_2009__45_2_373_0,
     author = {Rassoul-Agha, Firas and Sepp\"al\"ainen, Timo},
     title = {Almost sure functional central limit theorem for ballistic random walk in random environment},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {2},
     year = {2009},
     pages = {373-420},
     doi = {10.1214/08-AIHP167},
     zbl = {1176.60087},
     mrnumber = {2521407},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2009__45_2_373_0}
}
Rassoul-Agha, Firas; Seppäläinen, Timo. Almost sure functional central limit theorem for ballistic random walk in random environment. Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 2, pp. 373-420. doi : 10.1214/08-AIHP167. http://www.numdam.org/item/AIHPB_2009__45_2_373_0/

[1] N. Berger and O. Zeitouni. A quenched invariance principle for certain ballistic random walks in i.i.d. environments, 2008. Available at http://front.math.ucdavis.edu/math.PR/0702306. | MR 2477380 | Zbl 1173.82324

[2] E. Bolthausen and A.-S. Sznitman. On the static and dynamic points of view for certain random walks in random environment. Methods Appl. Anal. 9 (2002) 345-375. Special issue dedicated to Daniel W. Stroock and Srinivasa S. R. Varadhan on the occasion of their 60th birthday. | MR 2023130 | Zbl 1079.60079

[3] E. Bolthausen and A.-S. Sznitman. Ten Lectures on Random Media. Birkhäuser, Basel, 2002. | MR 1890289 | Zbl 1075.60128

[4] J. Bricmont and A. Kupiainen. Random walks in asymmetric random environments. Comm. Math. Phys. 142 (1991) 345-420. | MR 1137068 | Zbl 0734.60112

[5] D. L. Burkholder. Distribution function inequalities for martingales. Ann. Probability 1 (1973) 19-42. | MR 365692 | Zbl 0301.60035

[6] Y. Derriennic and M. Lin. The central limit theorem for Markov chains started at a point. Probab. Theory Related Fields 125 (2003) 73-76. | MR 1952457 | Zbl 1012.60028

[7] R. Durrett. Probability: Theory and Examples, 3rd edition. Brooks/Cole-Thomson, Belmont, CA, 2004. | MR 1068527 | Zbl 0709.60002

[8] S. N. Ethier and T. G. Kurtz. Markov Processes. Wiley, New York, 1986. | MR 838085 | Zbl 0592.60049

[9] W. Feller. An Introduction to Probability Theory and Its Applications. Vol. II, 2nd edition. Wiley, New York, 1971. | MR 270403 | Zbl 0219.60003

[10] I. Y. Goldsheid. Simple transient random walks in one-dimensional random environment: the central limit theorem. Probab. Theory Related Fields 139 (2007) 41-64. | MR 2322691 | Zbl 1134.60065

[11] M. Maxwell and M. Woodroofe. Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 (2000) 713-724. | MR 1782272 | Zbl 1044.60014

[12] F. Rassoul-Agha and T. Seppäläinen. An almost sure invariance principle for random walks in a space-time random environment. Probab. Theory Related Fields 133 (2005) 299-314. | MR 2198014 | Zbl 1088.60094

[13] F. Rassoul-Agha and T. Seppäläinen. Ballistic random walk in a random environment with a forbidden direction. ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006) 111-147 (electronic). | MR 2235176 | Zbl 1115.60106

[14] F. Rassoul-Agha and T. Seppäläinen. An almost sure invariance principle for ballistic random walks in product random environment, 2007. Available at http://front.math.ucdavis.edu/math.PR/0704.1022.

[15] F. Rassoul-Agha and T. Seppäläinen. Quenched invariance principle for multidimensional ballistic random walk in a random environment with a forbidden direction. Ann. Probab. 35 (2007) 1-31. | MR 2303942 | Zbl 1126.60090

[16] M. Rosenblatt. Markov Processes. Structure and Asymptotic Behavior. Springer, New York, 1971. | MR 329037 | Zbl 0236.60002

[17] F. Spitzer. Principles of Random Walks, 2nd edition. Springer, New York, 1976. | MR 388547 | Zbl 0359.60003

[18] A.-S. Sznitman. Slowdown estimates and central limit theorem for random walks in random environment. J. Eur. Math. Soc. (JEMS) 2 (2000) 93-143. | MR 1763302 | Zbl 0976.60097

[19] A.-S. Sznitman. An effective criterion for ballistic behavior of random walks in random environment. Probab. Theory Related Fields 122 (2002) 509-544. | MR 1902189 | Zbl 0995.60097

[20] A.-S. Sznitman. Topics in random walks in random environment. In School and Conference on Probability Theory. ICTP Lect. Notes, XVII. Abdus Salam Int. Cent. Theoret. Phys., Trieste (2004) 203-266 (electronic). | MR 2198849 | Zbl 1060.60102

[21] A.-S. Sznitman and O. Zeitouni. An invariance principle for isotropic diffusions in random environment. Invent. Math. 164 (2006) 455-567. | MR 2221130 | Zbl 1105.60079

[22] A.-S. Sznitman and M. Zerner. A law of large numbers for random walks in random environment. Ann. Probab. 27 (1999) 1851-1869. | MR 1742891 | Zbl 0965.60100

[23] O. Zeitouni. Random Walks in Random Environments. Springer, Berlin, 2004. | Zbl 1060.60103

[24] M. P. W. Zerner. Lyapounov exponents and quenched large deviations for multidimensional random walk in random environment. Ann. Probab. 26 (1998) 1446-1476. | MR 1675027 | Zbl 0937.60095