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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Abstract

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 $n^{-\gamma}$.

Résumé

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

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

Dates
First available in Project Euclid: 2 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1372772647

Digital Object Identifier
doi:10.1214/11-AIHP452

Mathematical Reviews number (MathSciNet)
MR3112437

Zentralblatt MATH identifier
1274.60144

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

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

Citation

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. https://projecteuclid.org/euclid.aihp/1372772647


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