Exit times for integrated random walks
Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 1, p. 167-193

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.

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.

DOI : https://doi.org/10.1214/13-AIHP577
Classification:  60G50,  60G40,  60F17
Keywords: 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},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {1},
     year = {2015},
     pages = {167-193},
     doi = {10.1214/13-AIHP577},
     zbl = {06412901},
     mrnumber = {3300967},
     language = {en},
     url = {http://www.numdam.org/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, Volume 51 (2015) no. 1, pp. 167-193. doi : 10.1214/13-AIHP577. http://www.numdam.org/item/AIHPB_2015__51_1_167_0/

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