Positivity of integrated random walks
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 1, p. 195-213

Take a centered random walk S n and consider the sequence of its partial sums A n := i=1 n S i . Suppose S 1 is in the domain of normal attraction of an α-stable law with 1<α2. Assuming that S 1 is either right-exponential (i.e. (S 1 >x|S 1 >0)=e -ax for some a>0 and all x>0) or right-continuous (skip free), we prove that {A 1 >0,,A N >0}C α N 1/(2α)-1/2 as N, where C α >0 depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.

Soit S n une marche aléatoire centrée, nous considérons la suite de ses sommes partielles A n := i=1 n S i . Nous supposons que S 1 est dans le domaine d’attraction normale d’une loi α-stable avec 1<α2. En supposant que S 1 est soit exponentielle à droite (i.e. (S 1 >x|S 1 >0)=e -ax ), soit continue à droite (i.e. (S 1 =1|S 1 >0)=1), nous prouvons que {A 1 >0,,A N >0}C α N 1/(2α)-1/2 quand N, où C α >0 dépend de la distribution de la marche. Nous considérons aussi une version conditionnelle de ce problème et nous étudions la positivité de ponts discrets intégrés.

DOI : https://doi.org/10.1214/12-AIHP487
Classification:  60G50,  60F99
Keywords: integrated random walk, persistence, one-sided exit probability, unilateral small deviations, area of random walk, Sparre-Andersen theorem, stable excursion, area of excursion
@article{AIHPB_2014__50_1_195_0,
     author = {Vysotsky, Vladislav},
     title = {Positivity of integrated random walks},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {1},
     year = {2014},
     pages = {195-213},
     doi = {10.1214/12-AIHP487},
     zbl = {1293.60053},
     mrnumber = {3161528},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_1_195_0}
}
Vysotsky, Vladislav. Positivity of integrated random walks. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 1, pp. 195-213. doi : 10.1214/12-AIHP487. http://www.numdam.org/item/AIHPB_2014__50_1_195_0/

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