Persistence of iterated partial sums
Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 3, pp. 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: 10.1214/11-AIHP452
Classification: 60G50, 60F10
Keywords: first passage time, iterated partial sums, persistence, lower tail probability, one-sided probability, random walk
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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://archive.numdam.org/articles/10.1214/11-AIHP452/

[1] F. Aurzada and C. Baumgarten. Survival probabilities of weighted random walks. ALEA Lat. Am. J. Probab. Math. Stat. 8 (2011) 235-258. | MR | Zbl

[2] F. Aurzada and S. Dereich. Universality of the asymptotics of the one-sided exit problem for integrated processes. Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013) 236-251. | Numdam | MR

[3] F. Caravenna and J.-D. Deuschel. Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction. Ann. Probab. 36 (2008) 2388-2433. | MR | Zbl

[4] A. Dembo, B. Poonen, Q. Shao and O. Zeitouni. Random polynomials having few or no real zeros. J. Amer. Math. Soc. 15 (2002) 857-892. | MR | Zbl

[5] A. Devulder, Z. Shi and T. Simon. The lower tail problem for the area of a symmetric stable process. Unpublished manuscript, 2007.

[6] W. Feller. An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edition. Wiley, New York, 1971. | MR | Zbl

[7] W. V. Li and Q.-M. Shao. Recent developments on lower tail probabilities for Gaussian processes. Cosmos 1 (2005) 95-106. | MR

[8] H. P. Mckean, Jr. A winding problem for a resonator driven by a white noise. J. Math. Kyoto Univ. 2 (1963) 227-235. | MR | Zbl

[9] S. J. Montgomery-Smith. Comparison of sums of independent identically distributed random vectors. Probab. Math. Statist. 14 (1993) 281-285. | MR | Zbl

[10] T. Simon. The lower tail problem for homogeneous functionals of stable processes with no negative jumps. ALEA Lat. Am. J. Probab. Math. Stat. 3 (2007) 165-179. | MR | Zbl

[11] Ya. G. Sinai. Distribution of some functionals of the integral of a random walk. Theoret. Math. Phys. 90 (1992) 219-241. | MR | Zbl

[12] V. Vysotsky. Clustering in a stochastic model of one-dimensional gas. Ann. Appl. Probab. 18 (2008) 1026-1058. | MR | Zbl

[13] V. Vysotsky. On the probability that integrated random walks stay positive. Stochastic Process. Appl. 120 (2010) 1178-1193. | MR | Zbl

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