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, Tome 46 (2010) no. 4, pp. 991-1000.

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.

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.

DOI : 10.1214/09-AIHP336
Classification : 60K37, 60F05
Mots-clés : Sinai's walk, internal DLA, random walks in random environments, excursion theory 
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     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
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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, Tome 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.

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