The Annals of Probability

A local limit theorem for random walks in random scenery and on randomly oriented lattices

Fabienne Castell, Nadine Guillotin-Plantard, Françoise Pène, and Bruno Schapira

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Abstract

Random walks in random scenery are processes defined by Zn := ∑k=1nξX1+⋯+Xk, where (Xk, k ≥ 1) and (ξy, y ∈ ℤ) are two independent sequences of i.i.d. random variables. We assume here that their distributions belong to the normal domain of attraction of stable laws with index α ∈ (0, 2] and β ∈ (0, 2], respectively. These processes were first studied by H. Kesten and F. Spitzer, who proved the convergence in distribution when α ≠ 1 and as n → ∞, of nδZn, for some suitable δ > 0 depending on α and β. Here, we are interested in the convergence, as n → ∞, of nδℙ(Zn = ⌊nδx⌋), when x ∈ ℝ is fixed. We also consider the case of random walks on randomly oriented lattices for which we obtain similar results.

Article information

Source
Ann. Probab., Volume 39, Number 6 (2011), 2079-2118.

Dates
First available in Project Euclid: 17 November 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1321539117

Digital Object Identifier
doi:10.1214/10-AOP606

Mathematical Reviews number (MathSciNet)
MR2932665

Zentralblatt MATH identifier
1238.60028

Subjects
Primary: 60F05: Central limit and other weak theorems 60G52: Stable processes

Keywords
Random walk in random scenery random walk on randomly oriented lattices local limit theorem stable process

Citation

Castell, Fabienne; Guillotin-Plantard, Nadine; Pène, Françoise; Schapira, Bruno. A local limit theorem for random walks in random scenery and on randomly oriented lattices. Ann. Probab. 39 (2011), no. 6, 2079--2118. doi:10.1214/10-AOP606. https://projecteuclid.org/euclid.aop/1321539117


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References

  • [1] Asselah, A. and Castell, F. (2007). Random walk in random scenery and self-intersection local times in dimensions d ≥ 5. Probab. Theory Related Fields 138 1–32.
  • [2] Bolthausen, E. (1989). A central limit theorem for two-dimensional random walks in random sceneries. Ann. Probab. 17 108–115.
  • [3] Borodin, A. N. (1979). A limit theorem for sums of independent random variables defined on a recurrent random walk. Dokl. Akad. Nauk SSSR 246 786–787.
  • [4] Borodin, A. N. (1979). Limit theorems for sums of independent random variables defined on a transient random walk. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 85 17–29, 237, 244.
  • [5] Borodin, A. N. (1984). Asymptotic behavior of local times of recurrent random walks with infinite variance. Teor. Veroyatnost. i Primenen. 29 312–326. Translation in Theory Probab. Appl. 29 (1984) 318–333.
  • [6] Campanino, M. and Petritis, D. (2003). Random walks on randomly oriented lattices. Markov Process. Related Fields 9 391–412.
  • [7] Castell, F. and Pradeilles, F. (2001). Annealed large deviations for diffusions in a random Gaussian shear flow drift. Stochastic Process. Appl. 94 171–197.
  • [8] Castell, F. (2004). Moderate deviations for diffusions in a random Gaussian shear flow drift. Ann. Inst. H. Poincaré Probab. Statist. 40 337–366.
  • [9] Chen, X. (2006). Moderate and small deviations for the ranges of one-dimensional random walks. J. Theoret. Probab. 19 721–739.
  • [10] Chen, X., Li, W. V. and Rosen, J. (2005). Large deviations for local times of stable processes and stable random walks in 1 dimension. Electron. J. Probab. 10 577–608 (electronic).
  • [11] Cohen, S. and Dombry, C. (2009). Convergence of dependent walks in a random scenery to fBm-local time fractional stable motions. J. Math. Kyoto Univ. 49 267–286.
  • [12] Csáki, E. and Révész, P. (1983). Strong invariance for local times. Z. Wahrsch. Verw. Gebiete 62 263–278.
  • [13] Csáki, E., König, W. and Shi, Z. (1999). An embedding for the Kesten–Spitzer random walk in random scenery. Stochastic Process. Appl. 82 283–292.
  • [14] den Hollander, F. and Steif, J. E. (2006). Random walk in random scenery: A survey of some recent results. In Dynamics & Stochastics. IMS Lecture Notes Monogr. Ser. 48 53–65. IMS, Beachwood, OH.
  • [15] Dombry, C. and Guillotin-Plantard, N. (2009). A functional approach for random walks in random sceneries. Electron. J. Probab. 14 1495–1512.
  • [16] Durrett, R. (1991). Probability: Theory and Examples. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA.
  • [17] Gantert, N., König, W. and Shi, Z. (2007). Annealed deviations of random walk in random scenery. Ann. Inst. H. Poincaré Probab. Statist. 43 47–76.
  • [18] Guillotin-Plantard, N. and Le Ny, A. (2008). A functional limit theorem for a 2D-random walk with dependent marginals. Electron. Commun. Probab. 13 337–351.
  • [19] Guillotin-Plantard, N. and Prieur, C. (2010). Limit theorem for random walk in weakly dependent random scenery. Ann. Inst. H. Poincaré Probab. Statist. 46 1178–1194.
  • [20] Gut, A. (2005). Probability: A Graduate Course. Springer, New York.
  • [21] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.
  • [22] Ibragimov, I. A. and Linnik, Y. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.
  • [23] Jain, N. C. and Pruitt, W. E. (1984). Asymptotic behavior of the local time of a recurrent random walk. Ann. Probab. 12 64–85.
  • [24] Kesten, H. and Spitzer, F. (1979). A limit theorem related to a new class of self-similar processes. Z. Wahrsch. Verw. Gebiete 50 5–25.
  • [25] Khoshnevisan, D. and Lewis, T. M. (1998). A law of the iterated logarithm for stable processes in random scenery. Stochastic Process. Appl. 74 89–121.
  • [26] Lacey, M. (1990). Large deviations for the maximum local time of stable Lévy processes. Ann. Probab. 18 1669–1675.
  • [27] Le Doussal, P. (1992). Diffusion in layered random flows, polymers, electrons in random potentials, and spin depolarization in random fields. J. Statist. Phys. 69 917–954.
  • [28] Le Gall, J.-F. and Rosen, J. (1991). The range of stable random walks. Ann. Probab. 19 650–705.
  • [29] Liggett, T. M. (1968). An invariance principle for conditioned sums of independent random variables. J. Math. Mech. 18 559–570.
  • [30] Matheron, G. and de Marsily, G. (1980). Is transport in porous media always diffusive? A counterxample. Water Resources Res. 16 901–907.
  • [31] Schmidt, K. (1984). On recurrence. Z. Wahrsch. Verw. Gebiete 68 75–95.
  • [32] Spitzer, F. (1964). Principles of Random Walk. Van Nostrand, Princeton, NJ.