Persistence of iterated partial sums
Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 3, p. 873-884

Let S n (2) denote the iterated partial sums. That is, S n (2) =S 1 +S 2 ++S n , where S i =X 1 +X 2 ++X i . Assuming X 1 ,X 2 ,...,X n are integrable, zero-mean, i.i.d. random variables, we show that the persistence probabilities p n (2) :=max 1in S i (2) < 0c𝔼|S n+1 | (n+1)𝔼|X 1 |, with c630 (and c=2 whenever X 1 is symmetric). The converse inequality holds whenever the non-zero min(-X 1 ,0) is bounded or when it has only finite third moment and in addition X 1 is squared integrable. Furthermore, p n (2) n -1/4 for any non-degenerate squared integrable, i.i.d., zero-mean X i . In contrast, we show that for any 0<γ<1/4 there exist integrable, zero-mean random variables for which the rate of decay of p n (2) is n -γ .

Soit S n (2) la somme partielle itérée, c’est à dire S n (2) =S 1 +S 2 ++S n , où S i =X 1 +X 2 ++X i . Pour des variables aléatoires X 1 ,X 2 ,...,X n i.i.d. intégrables et de moyenne nulle, nous montrons que les probabilités de persistance satisfont p n (2) :=max 1in S i (2) < 0c𝔼|S n+1 | (n+1)𝔼|X 1 |, avec c630 (et c=2 dès que X 1 est symétrique). En outre, l’inégalité inverse est vraie quand (-X 1 >t)e -αt pour un α>0 ou si (-X 1 >t) 1/t 0 quand t. Pour ces variables, on a donc p n (2) n -1/4 si X 1 admet un moment d’ordre 2. Par contre nous montrons que pour tout 0<γ<1/4, il existe des variables intégrables de moyenne nulle pour lesquelles p n (2) décroît comme n -γ .

DOI : https://doi.org/10.1214/11-AIHP452
Classification:  60G50,  60F10
Keywords: first passage time, iterated partial sums, persistence, lower tail probability, one-sided probability, random walk
@article{AIHPB_2013__49_3_873_0,
     author = {Dembo, Amir and Ding, Jian and Gao, Fuchang},
     title = {Persistence of iterated partial sums},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {49},
     number = {3},
     year = {2013},
     pages = {873-884},
     doi = {10.1214/11-AIHP452},
     zbl = {1274.60144},
     mrnumber = {3112437},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2013__49_3_873_0}
}
Dembo, Amir; Ding, Jian; Gao, Fuchang. Persistence of iterated partial sums. Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 3, pp. 873-884. doi : 10.1214/11-AIHP452. http://www.numdam.org/item/AIHPB_2013__49_3_873_0/

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