Random walks in random scenery are processes defined by , where and are two independent sequences of i.i.d. random variables with values in and respectively. We suppose that the distributions of and belong to the normal basin of attraction of stable distribution of index and . When and , 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 (i.e. or ) and . 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 (respectively for and ).
Les promenades aléatoires en paysage aléatoire sont des processus définis par , où et sont deux suites indépendantes de variables aléatoires i.i.d. à valeurs dans et respectivement. Nous supposons que les lois de et appartiennent au domaine d’attraction normal de lois stables d’indice et . Quand et , 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 (i.e. ou ) et . 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 (respectivement pour et ).
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] Scaling limit for trap models on . Ann. Probab. 35 (2007) 2356-2384. | MR | Zbl
and .[2] Convergence of Probability Measures. Wiley Series in Probability and Mathematical Statistics. Wiley, New York, 1968. | MR | Zbl
.[3] A central limit theorem for two-dimensional random walks in random sceneries. Ann. Probab. 17 (1989) 108-115. | MR | Zbl
.[4] 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] 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] 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] Moments and distribution of the local time of a two-dimensional random walk. Stochastic Process. Appl. 117 (2007) 262-270. | MR | Zbl
.[9] Moderate and small deviations for the ranges of one-dimensional random walks. J. Theor. Probab. 19 (2006) 721-739. | MR | Zbl
.[10] An asymptotic variance of the self-intersections of random walks. Sib. Math. J. 52 (2011) 639-650. | MR | Zbl
and .[11] 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
and .[12] An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edition. Wiley, New York, 1971. | MR | Zbl
.[13] On the dynamics of trap models in . Preprint. Available at arXiv:1010.5418.
and .[14] Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Lecture Notes in Mathematics 1766. Springer, Berlin, 2001. | MR | Zbl
and .[15] A limit theorem related to a new class of self-similar processes. Z. Wahrsch. Verw. Gebiete 50 (1979) 5-25. | MR | Zbl
and .[16] 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] The range of stable random walks. Ann. Probab. 19 (1991) 650-705. | MR | Zbl
and .[18] Convergence rates in the strong law for associated random variables. Probab. Math. Statist. 20 (2000) 203-214. | MR | Zbl
.[19] 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
and .[20] Is transport in porous media always diffusive? A counterxample. Water Resources Res. 16 (1980) 901-917.
and .[21] Principles of Random Walks. Van Nostrand, Princeton, NJ, 1964. | MR | Zbl
.[22] Limit theorems for stochastic processes. Theory Probab. Appl. 1 (1956) 261-290. | Zbl
.[23] Stochastic Process Limits. Springer Series in Operations Research. Springer, New York, 2002. | MR | Zbl
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