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.
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.
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}, 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://archive.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://archive.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://archive.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, Volume 51 (2015) no. 1, pp. 167-193. doi : 10.1214/13-AIHP577. http://archive.numdam.org/articles/10.1214/13-AIHP577/
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