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://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, Tome 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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