Exit times for integrated random walks

Denisov, Denis; Wachtel, Vitali

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015), p. 167-193 / Harvested from Numdam

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Résumé
Abstract

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 $n$ . Si le moment d’ordre $2 + δ$ est fini, nous montrons que cette probabilité décroit comme $n^{- 1 / 4}$ . 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 $n$ . Assuming that the moment of order $2 + δ$ is finite, we show that the exact asymptotics for this probability is $n^{- 1 / 4}$ . To show this asymptotics we develop a discrete potential theory for integrated random walks.

Publié le : 2015-01-01

MR 3300967 | Zbl 06412901

DOI : https://doi.org/10.1214/13-AIHP577
Classification: 60G50, 60G40, 60F17

@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},
     volume = {51},
     year = {2015},
     pages = {167-193},
     doi = {10.1214/13-AIHP577},
     mrnumber = {3300967},
     zbl = {06412901},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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) pp. 167-193. doi : 10.1214/13-AIHP577. http://gdmltest.u-ga.fr/item/AIHPB_2015__51_1_167_0/

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