Extremal points of high-dimensional random walks and mixing times of a brownian motion on the sphere

Eldan, Ronen

Eldan, Ronen

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014), p. 95-110 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé
Abstract

Nous étudions le comportement asymptotique de la probabilité que l’origine soit un point extrémal d’une marche aléatoire dans $ℝ^{n}$ . Nous montrons que cette probabilité est proche de $1 / 2$ si le nombre de pas de la marche aléatoire est entre $e^{n / (C log n)}$ et $e^{C n log n}$ pour une certaine constante $C > 0$ . Comme corollaire, nous obtenons une borne pour le temps de $\frac{π}{2}$ -recouvrement d’un mouvement brownien sphérique.

We derive asymptotics for the probability that the origin is an extremal point of a random walk in $ℝ^{n}$ . We show that in order for the probability to be roughly $1 / 2$ , the number of steps of the random walk should be between $e^{n / (C log n)}$ and $e^{C n log n}$ for some constant $C > 0$ . As a result, we attain a bound for the $\frac{π}{2}$ -covering time of a spherical Brownian motion.

Publié le : 2014-01-01

MR 3161524 | Zbl 1290.60079

DOI : https://doi.org/10.1214/12-AIHP515
Classification: 52A22, 52A38, 60J65

@article{AIHPB_2014__50_1_95_0,
     author = {Eldan, Ronen},
     title = {Extremal points of high-dimensional random walks and mixing times of a brownian motion on the sphere},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {50},
     year = {2014},
     pages = {95-110},
     doi = {10.1214/12-AIHP515},
     mrnumber = {3161524},
     zbl = {1290.60079},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2014__50_1_95_0}
}

Eldan, Ronen. Extremal points of high-dimensional random walks and mixing times of a brownian motion on the sphere. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) pp. 95-110. doi : 10.1214/12-AIHP515. http://gdmltest.u-ga.fr/item/AIHPB_2014__50_1_95_0/

Bibliographie

[1] D. J. Aldous. An introduction to covering problems for random walks on graphs. J. Theoret. Probab. 2 (1989) 87-89. | MR 981766 | Zbl 0684.60054

[2] D. Asimov. The grand tour: A tool for viewing multidimensional data. SIAM J. Sci. Statist. Comput. 6 (1985) 128-143. | MR 773286 | Zbl 0552.62052

[3] G. Baxter. A combinatorial lemma for complex numbers. Ann. Math. Statist. 32 (1961) 901-904. | MR 126290 | Zbl 0119.14201

[4] S. N. Bernstein. On certain modifications of Chebyshev's inequality. (Russian) Doklady Akademii Nauk SSSR 17(6) (1937) 275-277.

[5] A. Dembo, Y. Peres, J. Rosen and O. Zeitouni. Cover times for Brownian motion and random walks in two dimensions. Ann. of Math. (2) 160 (2004) 433-464. | MR 2123929 | Zbl 1068.60018

[6] P. Matthews. Covering problems for Brownian motion on spheres. Ann. Probab. 16(1) (1988) 189-199. | MR 920264 | Zbl 0638.60014

[7] P. Mörters and Y. Peres. Brownian Motion. Cambridge Univ. Press, Cambridge, 2010. | MR 2604525 | Zbl 1243.60002

[8] G. R. Shorack and J. A. Wellner. Empirical Processes with Applications to Statistics. Wiley, New York, 1986. | MR 838963 | Zbl 1170.62365

[9] J. G. Wendel. A problem in geometric probability. Math. Scand. 11 (1962) 109-111. | MR 146858 | Zbl 0108.31603