Limit theorems for one and two-dimensional random walks in random scenery
Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 2, pp. 506-528.

Random walks in random scenery are processes defined by Z n := k=1 n ξ X 1 ++X k , where (X k ,k1) and (ξ y ,y d ) are two independent sequences of i.i.d. random variables with values in d and respectively. We suppose that the distributions of X 1 and ξ 0 belong to the normal basin of attraction of stable distribution of index α(0,2] and β(0,2]. When d=1 and α1, a functional limit theorem has been established in (Z. Wahrsch. Verw. Gebiete 50 (1979) 5-25) and a local limit theorem in (Ann. Probab. To appear). In this paper, we establish the convergence in distribution and a local limit theorem when α=d (i.e. α=d=1 or α=d=2) and β(0,2]. Let us mention that functional limit theorems have been established in (Ann. Probab. 17 (1989) 108-115) and recently in (An asymptotic variance of the self-intersections of random walks. Preprint) in the particular case when β=2 (respectively for α=d=2 and α=d=1).

Les promenades aléatoires en paysage aléatoire sont des processus définis par Z n := k=1 n ξ X 1 ++X k , où (X k ,k1) et (ξ y ,y d ) sont deux suites indépendantes de variables aléatoires i.i.d. à valeurs dans d et respectivement. Nous supposons que les lois de X 1 et ξ 0 appartiennent au domaine d’attraction normal de lois stables d’indice α(0,2] et β(0,2]. Quand d=1 et α1, un théorème limite fonctionnel a été prouvé dans (Z. Wahrsch. Verw. Gebiete 50 (1979) 5-25) et un théorème limite local dans (Ann. Probab. To appear). Dans ce papier, nous prouvons la convergence en loi et un théorème limite local quand α=d (i.e. α=d=1 ou α=d=2) et β(0,2]. Mentionnons que des théorèmes limites fonctionnels ont été établis dans (Ann. Probab. 17 (1989) 108-115) et récemment dans (An asymptotic variance of the self-intersections of random walks. Preprint) dans le cas particulier où β=2 (respectivement pour α=d=2 et α=d=1).

DOI: 10.1214/11-AIHP466
Classification: 60F05, 60G52
Mots-clés : random walk in random scenery, local limit theorem, local time, stable process
@article{AIHPB_2013__49_2_506_0,
     author = {Castell, Fabienne and Guillotin-Plantard, Nadine and P\`ene, Fran\c{c}oise},
     title = {Limit theorems for one and two-dimensional random walks in random scenery},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {506--528},
     publisher = {Gauthier-Villars},
     volume = {49},
     number = {2},
     year = {2013},
     doi = {10.1214/11-AIHP466},
     mrnumber = {3088379},
     zbl = {1278.60046},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/11-AIHP466/}
}
TY  - JOUR
AU  - Castell, Fabienne
AU  - Guillotin-Plantard, Nadine
AU  - Pène, Françoise
TI  - Limit theorems for one and two-dimensional random walks in random scenery
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2013
SP  - 506
EP  - 528
VL  - 49
IS  - 2
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/11-AIHP466/
DO  - 10.1214/11-AIHP466
LA  - en
ID  - AIHPB_2013__49_2_506_0
ER  - 
%0 Journal Article
%A Castell, Fabienne
%A Guillotin-Plantard, Nadine
%A Pène, Françoise
%T Limit theorems for one and two-dimensional random walks in random scenery
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2013
%P 506-528
%V 49
%N 2
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/11-AIHP466/
%R 10.1214/11-AIHP466
%G en
%F AIHPB_2013__49_2_506_0
Castell, Fabienne; Guillotin-Plantard, Nadine; Pène, Françoise. Limit theorems for one and two-dimensional random walks in random scenery. Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 2, pp. 506-528. doi : 10.1214/11-AIHP466. http://www.numdam.org/articles/10.1214/11-AIHP466/

[1] G. Ben Arous and J. Cerný. Scaling limit for trap models on d . Ann. Probab. 35 (2007) 2356-2384. | MR | Zbl

[2] P. Billingsley. Convergence of Probability Measures. Wiley Series in Probability and Mathematical Statistics. Wiley, New York, 1968. | MR | Zbl

[3] E. Bolthausen. A central limit theorem for two-dimensional random walks in random sceneries. Ann. Probab. 17 (1989) 108-115. | MR | Zbl

[4] A. N. Borodin. A limit theorem for sums of independent random variables defined on a recurrent random walk. Dokl. Akad. Nauk SSSR 246 (1979) 786-787 (in Russian). | MR | Zbl

[5] A. N. Borodin. Limit theorems for sums of independent random variables defined on a transient random walk. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 85 (1979) 17-29, 237, 244. | MR | Zbl

[6] L. Breiman. Probability. Addison-Wesley, Reading, 1968. | MR | Zbl

[7] F. Castell, N. Guillotin-Plantard, F. Pène and Br. Schapira. A local limit theorem for random walks in random scenery and on randomly oriented lattices. Ann. Probab. 39 (2011) 2079-2118. | MR | Zbl

[8] J. Cerný. Moments and distribution of the local time of a two-dimensional random walk. Stochastic Process. Appl. 117 (2007) 262-270. | MR | Zbl

[9] X. Chen. Moderate and small deviations for the ranges of one-dimensional random walks. J. Theor. Probab. 19 (2006) 721-739. | MR | Zbl

[10] G. Deligiannidis and S. Utev. An asymptotic variance of the self-intersections of random walks. Sib. Math. J. 52 (2011) 639-650. | MR | Zbl

[11] A. Dvoretzky and P. Erdös. Some problems on random walk in space. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950 353-367. Univ. California Press, Berkeley, CA, 1950. | MR | Zbl

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

[13] L. R. Fontes and P. Mathieu. On the dynamics of trap models in d . Preprint. Available at arXiv:1010.5418.

[14] H. Hennion and L. Hervé. Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Lecture Notes in Mathematics 1766. Springer, Berlin, 2001. | MR | Zbl

[15] H. Kesten and F. Spitzer. A limit theorem related to a new class of self-similar processes. Z. Wahrsch. Verw. Gebiete 50 (1979) 5-25. | MR | Zbl

[16] P. Le Doussal. Diffusion in layered random flows, polymers, electrons in random potentials, and spin depolarization in random fields. J. Statist. Phys. 69 (1992) 917-954. | MR | Zbl

[17] J. F. Le Gall and J. Rosen. The range of stable random walks. Ann. Probab. 19 (1991) 650-705. | MR | Zbl

[18] S. Louhichi. Convergence rates in the strong law for associated random variables. Probab. Math. Statist. 20 (2000) 203-214. | MR | Zbl

[19] S. Louhichi and E. Rio. Convergence du processus de sommes partielles vers un processus de Lévy pour les suites associées. C. R. Acad. Sci. Paris 349 (2011) 89-91. | MR | Zbl

[20] G. Matheron and G. De Marsily. Is transport in porous media always diffusive? A counterxample. Water Resources Res. 16 (1980) 901-917.

[21] F. Spitzer. Principles of Random Walks. Van Nostrand, Princeton, NJ, 1964. | MR | Zbl

[22] A. V. Skorokhod. Limit theorems for stochastic processes. Theory Probab. Appl. 1 (1956) 261-290. | Zbl

[23] W. Whitt. Stochastic Process Limits. Springer Series in Operations Research. Springer, New York, 2002. | MR | Zbl

Cited by Sources: