Abstract
LetI n denote the number of common points to the paths, up to timen, of two independent random walks with values in ℤ4. The sequence (logn)−1 I n is shown to converge in distribution towards the square of a normal variable. Limit theorems are also proved for some processes related to the sequence (I n ), which lead to a better understanding of recent results obtained by G.F. Lawler. Similar statements are proved for the paths of three independent random walks with values in ℤ3.
Similar content being viewed by others
Bibliographie
Aizenman, M: Geometric analysis of ø4 fields and Ising models. Commun. Math. Phys.86, 1–48 (1982)
Erdös, P., Taylor, S.J.: Some intersection properties of random walk paths. Acta Math. Acad. Sci. Hung.11, 231–248 (1960)
Felder, G., Fröhlich, J.: Intersection properties of simple random walks: a renormalization group approach. Commun. Math. Phys.97, 111–124 (1985)
Geman, D., Horowitz, J., Rosen, J.: A local time analysis of intersections of Brownian paths in the plane. Ann. Probab.12, 86–107 (1984)
Kasahara, Y., Kotani, S.: On limit processes for a class of additive functionals of recurrent diffusion processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.49, 133–153 (1979)
Lawler, G.F.: The probability of intersection of independent random walks in four dimensions. Commun. Math. Phys.86, 539–554 (1982)
Lawler, G.F.: Intersections of random walks in four dimensions. II. Commun. Math. Phys.97, 583–594 (1985)
Lawler, G.F.: The probability of intersection of three independent random walks in three dimensions (preprint)
Le Gall, J.-F.: Sur le temps local d'intersection du mouvement brownien plan et la méthode de renormalisation de Varadhan. Séminaire de Probab. XIX. Lecture Notes in Mathematics, Vol. 1123. Berlin, Heidelberg, New York: Springer 1985, pp. 314–331
Le Gall, J.-F.: Sur la saucisse de Wiener et les points multiples du mouvement brownien (preprint, à paraître dans Ann. Probab.)
Le Gall, J.-F.: Propriétés d'intersection des marches aléatoires. I. Convergence vers le temps local d'intersection. Commun. Math. Phys.104, 467–503 (1986)
Spitzer, F.: Principles of random walk. Princeton, NJ: Van Nostrand 1964
Skorokhod, A.V.: Limit theorems for stochastic processes. Theory Probab. Appl.1, 261–290 (1956)
Yor, M.: Renormalisation et convergence en loi pour les temps locaux d'intersection du mouvement brownien dans ℝ3. Séminaire de Probab. XIX. Lecture Notes in Mathematics, Vol. 1123. Berlin, Heidelberg, New York: Springer 1985, pp. 350–365
Yor, M.: Renormalisation et convergence en loi pour certaines intégrales multiples associées au mouvement brownien dans ℝd (à paraître)
Author information
Authors and Affiliations
Additional information
Communicated by J. Fröhlich
Rights and permissions
About this article
Cite this article
Le Gall, J.F. Propriétés d'intersection des marches aléatoires. Commun.Math. Phys. 104, 509–528 (1986). https://doi.org/10.1007/BF01210953
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01210953