A functional approach for random walks in random sceneries

Clement Dombry (Universite de Poitiers)
Nadine Guillotin-Plantard (Université Lyon Claude Bernard)

Abstract


A functional approach for the study of the random walks in random sceneries (RWRS) is proposed. Under fairly general assumptions on the random walk and on the random scenery, functional limit theorems are proved. The method allows to study separately the convergence of the walk and of the scenery: on the one hand, a general criterion for the convergence of the local time of the walk is provided, on the other hand, the convergence of the random measures associated with the scenery is studied. This functional approach is robust enough to recover many of the known results on RWRS as well as new ones, including the case of many walkers evolving in the same scenery.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1495-1512

Publication Date: July 2, 2009

DOI: 10.1214/EJP.v14-659

References

  1. Billingsley, Patrick. Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp. MR0233396 (38 #1718)

  2. Bolthausen, Erwin. A central limit theorem for two-dimensional random walks in random sceneries. Ann. Probab. 17 (1989), no. 1, 108--115. MR0972774 (90h:60020)

  3. Borodin, A. N. Limit theorems for sums of independent random variables defined on a transient random walk. (Russian) Investigations in the theory of probability distributions, IV. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 85 (1979), 17--29, 237, 244. MR0535455 (80j:60029)

  4. Borodin, A. N. Limit theorems for sums of independent random variables defined on a recurrent random walk. (Russian) Teor. Veroyatnost. i Primenen. 28 (1983), no. 1, 98--114. MR0691470 (84g:60033)

  5. Cadre, B. Etude de convergence en loi de fonctionnelles de processus: Formes quadratiques ou multilinéaires aléatoires, Temps locaux d'intersection de marches aléatoires, Théorème central limite presque sûr. (1995) PHD Thesis, Université Rennes 1.

  6. Chen, X. ; Khoshnevisan, D. From charged polymers to random walk in random scenery. (2009) To appear in Proceedings of the Third Erich L. Lehmann Symposium.

  7. Chen, Xia; Li, Wenbo V. Large and moderate deviations for intersection local times. Probab. Theory Related Fields 128 (2004), no. 2, 213--254. MR2031226 (2005m:60175)

  8. Cohen, S.; Dombry, C. Convergence of dependent walks in a random scenery to fBm-local time fractional stable motions. (2009) To appear in Journal of Mathematics of Kyoto University.

  9. Cohen, Serge; Samorodnitsky, Gennady. Random rewards, fractional Brownian local times and stable self-similar processes. Ann. Appl. Probab. 16 (2006), no. 3, 1432--1461. MR2260069 (2008b:60080)

  10. Dobrushin, R. L.; Major, P. Non-central limit theorems for nonlinear functionals of Gaussian fields. Z. Wahrsch. Verw. Gebiete 50 (1979), no. 1, 27--52. MR0550122 (81i:60019)

  11. Dombry,C. Convergence to stable noise and applications. (2009) Preprint.

  12. Dombry, C.; Guillotin-Plantard, N. Discrete approximation of a stable self-similar stationary increments process. (2009) Bernoulli, Vol. 15, No 1, 195--222.

  13. Guillotin-Plantard, N.; Le Ny, A. Transient random walks on 2d-oriented lattices. (2007) Theory of Probability and Its Applications (TVP), Vol. 52, No 4, 815—826.

  14. Guillotin-Plantard, Nadine; Le Ny, Arnaud. A functional limit theorem for a 2D-random walk with dependent marginals. Electron. Commun. Probab. 13 (2008), 337--351. MR2415142 (2009e:60079)

  15. Guillotin-Plantard, N.; Prieur, C. Central limit theorem for sampled sums of dependent random variables. (2009) To appear in ESAIM P\&S.

  16. Guillotin-Plantard, N.; Prieur, C. Limit theorem for random walk in weakly dependent random scenery. (2009) Preprint.

  17. Kesten, H.; Spitzer, F. A limit theorem related to a new class of self-similar processes. Z. Wahrsch. Verw. Gebiete 50 (1979), no. 1, 5--25. MR0550121 (82a:60149)

  18. Lang, Reinhard; Nguyen, Xuan-Xanh. Strongly correlated random fields as observed by a random walker. Z. Wahrsch. Verw. Gebiete 64 (1983), no. 3, 327--340. MR0716490 (86a:60046)

  19. Le Borgne, Stéphane. Exemples de systèmes dynamiques quasi-hyperboliques à décorrélations lentes. (French) [Examples of quasi-hyperbolic dynamical systems with slow decay of correlations] C. R. Math. Acad. Sci. Paris 343 (2006), no. 2, 125--128. MR2243306 (2007f:37041)

  20. Ledoux, Michel; Talagrand, Michel. Probability in Banach spaces. Isoperimetry and processes. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 23. Springer-Verlag, Berlin, 1991. xii+480 pp. ISBN: 3-540-52013-9 MR1102015 (93c:60001)

  21. Maejima, Makoto. Limit theorems related to a class of operator-self-similar processes. Nagoya Math. J. 142 (1996), 161--181. MR1399472 (97g:60033)

  22. Nualart, David. Stochastic integration with respect to fractional Brownian motion and applications. Stochastic models (Mexico City, 2002), 3--39, Contemp. Math., 336, Amer. Math. Soc., Providence, RI, 2003. MR2037156 (2004m:60119)

  23. Nualart, David. The Malliavin calculus and related topics. Second edition. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. xiv+382 pp. ISBN: 978-3-540-28328-7; 3-540-28328-5 MR2200233 (2006j:60004) 

  24. Pène, F. Transient random walk in $Z^2$ with stationary orientations. (2007) To appear in ESAIM P&S.

  25. Pipiras, Vladas; Taqqu, Murad S. Integration questions related to fractional Brownian motion. Probab. Theory Related Fields 118 (2000), no. 2, 251--291. MR1790083 (2002c:60091)

  26. Wang, Wensheng. Weak convergence to fractional Brownian motion in Brownian scenery. Probab. Theory Related Fields 126 (2003), no. 2, 203--220. MR1990054 (2004d:60265)

Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.