Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Persistence of iterated partial sums

Amir Dembo, Jian Ding, and Fuchang Gao

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Let $S_{n}^{(2)}$ denote the iterated partial sums. That is, $S_{n}^{(2)}=S_{1}+S_{2}+\cdots+S_{n}$, where $S_{i}=X_{1}+X_{2}+\cdots+X_{i}$. Assuming $X_{1},X_{2},\ldots,X_{n}$ are integrable, zero-mean, i.i.d. random variables, we show that the persistence probabilities

\[p_{n}^{(2)}:=\mathbb{P}\Bigl(\max_{1\le i\le n}S_{i}^{(2)}<0\Bigr)\le c\sqrt{\frac{\mathbb{E}|S_{n+1}|}{(n+1)\mathbb{E}|X_{1}|}},\]

with $c\le6\sqrt{30}$ (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, $p_{n}^{(2)}\asymp n^{-1/4}$ for any non-degenerate squared integrable, i.i.d., zero-mean $X_{i}$. In contrast, we show that for any $0<\gamma<1/4$ there exist integrable, zero-mean random variables for which the rate of decay of $p_{n}^{(2)}$ is $n^{-\gamma}$.


Soit $S_{n}^{(2)}$ la somme partielle itérée, c’est à dire $S_{n}^{(2)}=S_{1}+S_{2}+\cdots+S_{n}$, où $S_{i}=X_{1}+X_{2}+\cdots+X_{i}$. Pour des variables aléatoires $X_{1},X_{2},\ldots,X_{n}$ i.i.d. intégrables et de moyenne nulle, nous montrons que les probabilités de persistance satisfont

\[p_{n}^{(2)}:=\mathbb{P}\Bigl(\max_{1\le i\le n}S_{i}^{(2)}<0\Bigr)\le c\sqrt{\frac{\mathbb{E}|S_{n+1}|}{(n+1)\mathbb{E}|X_{1}|}},\]

avec $c\le6\sqrt{30}$ (et $c=2$ dès que $X_{1}$ est symétrique). En outre, l’inégalité inverse est vraie quand $\mathbb{P}(-X_{1}>t)\asymp e^{-\alpha t}$ pour un $\alpha>0$ ou si $\mathbb{P}(-X_{1}>t)^{1/t}\to0$ quand $t\to\infty$. Pour ces variables, on a donc $p_{n}^{(2)}\asymp n^{-1/4}$ si $X_{1}$ admet un moment d’ordre 2. Par contre nous montrons que pour tout $0<\gamma<1/4$, il existe des variables intégrables de moyenne nulle pour lesquelles $p_{n}^{(2)}$ décroît comme $n^{-\gamma}$.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 49, Number 3 (2013), 873-884.

First available in Project Euclid: 2 July 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks 60F10: Large deviations

First passage time Iterated partial sums Persistence Lower tail probability One-sided probability Random walk


Dembo, Amir; Ding, Jian; Gao, Fuchang. Persistence of iterated partial sums. Ann. Inst. H. Poincaré Probab. Statist. 49 (2013), no. 3, 873--884. doi:10.1214/11-AIHP452.

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  • [1] F. Aurzada and C. Baumgarten. Survival probabilities of weighted random walks. ALEA Lat. Am. J. Probab. Math. Stat. 8 (2011) 235–258.
  • [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.
  • [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.
  • [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.
  • [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.
  • [7] W. V. Li and Q.-M. Shao. Recent developments on lower tail probabilities for Gaussian processes. Cosmos 1 (2005) 95–106.
  • [8] H. P. McKean, Jr. A winding problem for a resonator driven by a white noise. J. Math. Kyoto Univ. 2 (1963) 227–235.
  • [9] S. J. Montgomery-Smith. Comparison of sums of independent identically distributed random vectors. Probab. Math. Statist. 14 (1993) 281–285.
  • [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.
  • [11] Ya. G. Sinai. Distribution of some functionals of the integral of a random walk. Theoret. Math. Phys. 90 (1992) 219–241.
  • [12] V. Vysotsky. Clustering in a stochastic model of one-dimensional gas. Ann. Appl. Probab. 18 (2008) 1026–1058.
  • [13] V. Vysotsky. On the probability that integrated random walks stay positive. Stochastic Process. Appl. 120 (2010) 1178–1193.