Nous considérons une marche aléatoire centrée de variance finie et étudions le comportement asymptotique de la probabilité que l’aire sous la marche reste positive jusqu’à un grand temps . Si le moment d’ordre est fini, nous montrons que cette probabilité décroit comme . Pour prouver ce comportement asymptotique, nous développons une théorie du potentiel discrète pour des marches aléatoires intégrées.
We consider a centered random walk with finite variance and investigate the asymptotic behaviour of the probability that the area under this walk remains positive up to a large time . Assuming that the moment of order is finite, we show that the exact asymptotics for this probability is . To show this asymptotics we develop a discrete potential theory for integrated random walks.
Mots clés : Markov chain, exit time, harmonic function, Weyl chamber, normal approximation, Kolmogorov diffusion
@article{AIHPB_2015__51_1_167_0, author = {Denisov, Denis and Wachtel, Vitali}, title = {Exit times for integrated random walks}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {167--193}, publisher = {Gauthier-Villars}, volume = {51}, number = {1}, year = {2015}, doi = {10.1214/13-AIHP577}, mrnumber = {3300967}, zbl = {1310.60049}, language = {en}, url = {http://www.numdam.org/articles/10.1214/13-AIHP577/} }
TY - JOUR AU - Denisov, Denis AU - Wachtel, Vitali TI - Exit times for integrated random walks JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2015 SP - 167 EP - 193 VL - 51 IS - 1 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/13-AIHP577/ DO - 10.1214/13-AIHP577 LA - en ID - AIHPB_2015__51_1_167_0 ER -
%0 Journal Article %A Denisov, Denis %A Wachtel, Vitali %T Exit times for integrated random walks %J Annales de l'I.H.P. Probabilités et statistiques %D 2015 %P 167-193 %V 51 %N 1 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/13-AIHP577/ %R 10.1214/13-AIHP577 %G en %F AIHPB_2015__51_1_167_0
Denisov, Denis; Wachtel, Vitali. Exit times for integrated random walks. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 167-193. doi : 10.1214/13-AIHP577. http://www.numdam.org/articles/10.1214/13-AIHP577/
[1] M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions Formulas, Graphs and Mathematical Tables. Dover Publications Inc., New York. 1992. Reprint of the 1972 edition. | MR | Zbl
[2] Universality of asymptotics of the one-sided exit problem for integrated processes. Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013) 236–251. | DOI | Numdam | MR | Zbl
and .[3] Persistence probabilities for an integrated random walk bridge. Available at arXiv:1205.2895, 2012. | MR | Zbl
, and .[4] Pinning and wetting transition in -dimensional fields with Laplacian interaction. Ann. Probab. 36 (2008) 2388–2433. | DOI | MR | Zbl
and .[5] Persistence of iterated partial sums. Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013) 873–884. Available at arXiv:1205.5596. | DOI | Numdam | MR | Zbl
, and .[6] Conditional limit theorems for ordered random walks. Electron. J. Probab. 15 (2010) 292–322. | DOI | MR | Zbl
and .[7] Random walks in cones. Ann. Probab. To appear. Available at arXiv:1110.1254, 2011. | MR | Zbl
and .[8] Bounds on the concentration function in terms of the Diophantine approximation. C. R. Math. Acad. Sci. Paris 345 (2007) 513–518. | DOI | MR | Zbl
and .[9] Integrated Brownian motion, conditioned to be positive. Ann. Probab. 27 (1999) 1283–1303. | DOI | MR | Zbl
, and .[10] Sur les excursions de l’intégrale du mouvement brownien. C. R. Acad. Sci. Paris Sér. I Math. 314 (1992) 1053–1056. | MR | Zbl
.[11] The approximation of partial sums of RV’s. Z. Wahrsch. verw. Gebiete 35 (1976) 213–220. | MR | Zbl
.[12] Jr. A winding problem for a resonator driven by a white noise. J. Math. Kyoto Univ. 2 (1963) 227–235. | MR | Zbl
.[13] Distribution of some functionals of the integral of a random walk. Theor. Math. Phys. 90 (1992) 219–241. | DOI | MR | Zbl
.[14] On the probability that integrated random walks stay positive. Stochastic Process. Appl. 120 (2010) 1178–1193. | DOI | MR | Zbl
.[15] Positivity of integrated random walks. Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014) 195–213. | DOI | Numdam | MR | Zbl
.Cité par Sources :