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 ${Z}_{n}:={\sum }_{k=1}^{n}{\xi }_{{X}_{1}+\cdots +{X}_{k}}$, where $\left({X}_{k},k\ge 1\right)$ and $\left({\xi }_{y},y\in {ℤ}^{d}\right)$ are two independent sequences of i.i.d. random variables with values in ${ℤ}^{d}$ and $ℝ$ respectively. We suppose that the distributions of ${X}_{1}$ and ${\xi }_{0}$ belong to the normal basin of attraction of stable distribution of index $\alpha \in \left(0,2\right]$ and $\beta \in \left(0,2\right]$. When $d=1$ and $\alpha \ne 1$, 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 $\alpha =d$ (i.e. $\alpha =d=1$ or $\alpha =d=2$) and $\beta \in \left(0,2\right]$. 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 $\beta =2$ (respectively for $\alpha =d=2$ and $\alpha =d=1$).

Les promenades aléatoires en paysage aléatoire sont des processus définis par ${Z}_{n}:={\sum }_{k=1}^{n}{\xi }_{{X}_{1}+\cdots +{X}_{k}}$, où $\left({X}_{k},k\ge 1\right)$ et $\left({\xi }_{y},y\in {ℤ}^{d}\right)$ sont deux suites indépendantes de variables aléatoires i.i.d. à valeurs dans ${ℤ}^{d}$ et $ℝ$ respectivement. Nous supposons que les lois de ${X}_{1}$ et ${\xi }_{0}$ appartiennent au domaine d’attraction normal de lois stables d’indice $\alpha \in \left(0,2\right]$ et $\beta \in \left(0,2\right]$. Quand $d=1$ et $\alpha \ne 1$, 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 $\alpha =d$ (i.e. $\alpha =d=1$ ou $\alpha =d=2$) et $\beta \in \left(0,2\right]$. 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 $\beta =2$ (respectivement pour $\alpha =d=2$ et $\alpha =d=1$).

DOI: 10.1214/11-AIHP466
Classification: 60F05,  60G52
Keywords: random walk in random scenery, local limit theorem, local time, stable process
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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 ${ℤ}^{d}$. 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

[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 ${ℤ}^{d}$. 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

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