A functional limit theorem for a 2d-random walk with dependent marginals

Nadine Guillotin-Plantard (Université Lyon 1)
Arnaud Le Ny (Université Paris Sud)


We prove a non-standard functional limit theorem for a two dimensional simple random walk on some randomly oriented lattices. This random walk, already known to be transient, has different horizontal and vertical fluctuations leading to different normalizations in the functional limit theorem, with a non-Gaussian horizontal behavior. We also prove that the horizontal and vertical components are not asymptotically independent.

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Pages: 337-351

Publication Date: June 20, 2008

DOI: 10.1214/ECP.v13-1386


  1. P. Billingsley. Convergence of probability measures. 2nd edition, Wiley, New York, (1999). MR1700749
  2. M. Campanino and D. Pétritis. On the physical relevance of random walks: an example of random walks on randomly oriented lattices "Random walks and geometry", V. Kaimanovitch (ed.), Walter de Gruyter (2004), 393--411. MR2087791
  3. M. Campanino and D. Pétritis. Random walks on randomly oriented lattices. Mark. Proc. Rel. Fields 9 (2003), 391-412. MR2028220
  4. C. Dombry and N. Guillotin-Plantard. Discrete approximation of a stable self-similar stationary increments process. To appear in Bernoulli (2008).
  5. N. Guillotin-Plantard and A. Le Ny. Transient random walks in dimension two. Theo. Probab. Appl. 52, No 4 (2007), 815--826.
  6. 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. MR0550121
  7. J.F. Le Gall. Mouvement Brownien, processus de branchement et superprocessus. Notes de Cours de DEA, Ecole Normale SupÈrieure. Available on the website of E.N.S., rue d'Ulm, dÈpartement de mathÈmatiques (1994).
  8. D. Revuz and M. Yor. Continuous martingales and Brownian motion. Springer (1991). MR1083357

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