no. 4
The Arcsine law as the limit of the internal DLA cluster generated by Sinai's walk
Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 4, pp. 991-1000.

We identify the limit of the internal DLA cluster generated by Sinai's walk as the law of a functional of a brownian motion which turns out to be a new interpretation of the Arcsine law.

On détermine la loi limite du cluster de diffusion à agrégation limitée interne comme celle d'une fonctionnelle du mouvement brownien, qui donne une nouvelle interprétation de la loi de l'Arcsinus.

DOI: 10.1214/09-AIHP336
Classification: 60K37, 60F05
Keywords: Sinai's walk, internal DLA, random walks in random environments, excursion theory 
@article{AIHPB_2010__46_4_991_0,
     author = {Enriquez, N. and Lucas, C. and Simenhaus, F.},
     title = {The {Arcsine} law as the limit of the internal {DLA} cluster generated by {Sinai's} walk},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {991--1000},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {4},
     year = {2010},
     doi = {10.1214/09-AIHP336},
     mrnumber = {2744882},
     zbl = {1210.82028},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/09-AIHP336/}
}
TY  - JOUR
AU  - Enriquez, N.
AU  - Lucas, C.
AU  - Simenhaus, F.
TI  - The Arcsine law as the limit of the internal DLA cluster generated by Sinai's walk
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2010
SP  - 991
EP  - 1000
VL  - 46
IS  - 4
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/09-AIHP336/
DO  - 10.1214/09-AIHP336
LA  - en
ID  - AIHPB_2010__46_4_991_0
ER  - 
%0 Journal Article
%A Enriquez, N.
%A Lucas, C.
%A Simenhaus, F.
%T The Arcsine law as the limit of the internal DLA cluster generated by Sinai's walk
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2010
%P 991-1000
%V 46
%N 4
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/09-AIHP336/
%R 10.1214/09-AIHP336
%G en
%F AIHPB_2010__46_4_991_0
Enriquez, N.; Lucas, C.; Simenhaus, F. The Arcsine law as the limit of the internal DLA cluster generated by Sinai's walk. Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 4, pp. 991-1000. doi : 10.1214/09-AIHP336. http://archive.numdam.org/articles/10.1214/09-AIHP336/

[1] J. Bertoin. Subordinators: Examples and applications. In Lectures on Probability Theory and Statistics (Saint-Flour, 1997). Lecture Notes in Math. 1717 1-91. Springer, Berlin, 1999. | MR | Zbl

[2] P. Diaconis and W. Fulton. A growth model, a game, an algebra, Lagrange inversion, and characteristic classes. Rend. Sem. Mat. Univ. Politec. Torino 49 (1993) 95-119. | MR | Zbl

[3] P. G. Hoel, S. C. Port and C. J. Stone. Introduction to Stochastic Processes. Houghton Mifflin, Boston, MA, 1972. | MR | Zbl

[4] G. F. Lawler, M. Bramson and D. Griffeath. Internal diffusion limited aggregation. Ann. Probab. 20 (1992) 2117-2140. | MR | Zbl

[5] P. Lévy. Sur certains processus stochastiques homogènes. Compos. Math. 7 (1939) 283-339. | Numdam | MR | Zbl

[6] R. Mansuy and M. Yor. Aspects of Brownian Motion. Springer, Berlin, 2008. | MR | Zbl

[7] D. Revuz and M. Yor. Continuous martingales and Brownian Motion, 3rd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin, 1999. | MR | Zbl

[8] Y. G. Sinai. The limit behavior of a one-dimensional random walk in a random environment. Teor. Veroyatn. Primen. 27 (1982) 247-258. | MR | Zbl

[9] S. Tavaré and O. Zeitouni. Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1837. Springer, Berlin, 2004. | MR | Zbl

[10] M. Yor. Local Times and Excursions for Brownian Motion: A Concise Introduction Lecciones en Mathematicas. Universidad Central de Venezuela, Caracas, 1995.

Cited by Sources: