Persistence of iterated partial sums

Dembo, Amir; Ding, Jian; Gao, Fuchang

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 3, pp. 873-884

Let $S_n^{\left(2\right)}$ denote the iterated partial sums. That is, $S_n^{\left(2\right)}=S_1+S_2+\cdots +S_n$, where $S_i=X_1+X_2+\cdots +X_i$. Assuming $X_1,X_2,...,X_n$ are integrable, zero-mean, i.i.d. random variables, we show that the persistence probabilities

$$p_n^{\left(2\right)}:=\mathbb{P}\left(\underset{1\le i\le n}{max}{S}_i^{\left(2\right)}<0\right)\le c\sqrt{\frac{𝔼|S_{n+1}|}{(n+1)𝔼|X_1|}},$$

with $c\le 6\sqrt{30}$ (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, $p_n^{\left(2\right)}\asymp n^{-1/4}$ 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 $p_n^{\left(2\right)}$ is $n^{-\gamma }$.

Soit $S_n^{\left(2\right)}$ la somme partielle itérée, c’est à dire $S_n^{\left(2\right)}=S_1+S_2+\cdots +S_n$, où $S_i=X_1+X_2+\cdots +X_i$. Pour des variables aléatoires $X_1,X_2,...,X_n$ i.i.d. intégrables et de moyenne nulle, nous montrons que les probabilités de persistance satisfont

$$p_n^{\left(2\right)}:=\mathbb{P}\left(\underset{1\le i\le n}{max}{S}_i^{\left(2\right)}<0\right)\le c\sqrt{\frac{𝔼|S_{n+1}|}{(n+1)𝔼|X_1|}},$$

avec $c\le 6\sqrt{30}$ (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 $\mathbb{P}{(-X_1>t)}^{1/t}\to 0$ quand $t\to \infty$. Pour ces variables, on a donc $p_n^{\left(2\right)}\asymp n^{-1/4}$ 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 $p_n^{\left(2\right)}$ décroît comme $n^{-\gamma }$.

Classification : 60G50, 60F10
Keywords: first passage time, iterated partial sums, persistence, lower tail probability, one-sided probability, random walk

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

[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. | MR | Numdam

[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. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

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

[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. | Zbl | MR

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

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

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