Open Access
February 2014 Positivity of integrated random walks
Vladislav Vysotsky
Ann. Inst. H. Poincaré Probab. Statist. 50(1): 195-213 (February 2014). DOI: 10.1214/12-AIHP487


Take a centered random walk $S_{n}$ and consider the sequence of its partial sums $A_{n}:=\sum_{i=1}^{n}S_{i}$. Suppose $S_{1}$ is in the domain of normal attraction of an $\alpha$-stable law with $1<\alpha\le2$. Assuming that $S_{1}$ is either right-exponential (i.e. $\mathbb{P}(S_{1}>x|S_{1}>0)=\mathrm{e}^{-ax}$ for some $a>0$ and all $x>0$) or right-continuous (skip free), we prove that

\[\mathbb{P}\{A_{1}>0,\dots,A_{N}>0\}\sim C_{\alpha}N^{{1}/{(2\alpha)}-1/2}\]

as $N\to\infty$, where $C_{\alpha}>0$ depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.

Soit $S_{n}$ une marche aléatoire centrée, nous considérons la suite de ses sommes partielles $A_{n}:=\sum_{i=1}^{n}S_{i}$. Nous supposons que $S_{1}$ est dans le domaine d’attraction normale d’une loi $\alpha$-stable avec $1<\alpha\le2$. En supposant que $S_{1}$ est soit exponentielle à droite (i.e. $\mathbb{P}(S_{1}>x|S_{1}>0)=\mathrm{e}^{-ax}$), soit continue à droite (i.e. $\mathbb{P}(S_{1}=1|S_{1}>0)=1$), nous prouvons que

\[\mathbb{P}\{A_{1}>0,\dots,A_{N}>0\}\sim C_{\alpha}N^{{1}/{(2\alpha)}-1/2}\]

quand $N\to\infty$, où $C_{\alpha}>0$ dépend de la distribution de la marche. Nous considérons aussi une version conditionnelle de ce problème et nous étudions la positivité de ponts discrets intégrés.


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Vladislav Vysotsky. "Positivity of integrated random walks." Ann. Inst. H. Poincaré Probab. Statist. 50 (1) 195 - 213, February 2014.


Published: February 2014
First available in Project Euclid: 1 January 2014

zbMATH: 1293.60053
MathSciNet: MR3161528
Digital Object Identifier: 10.1214/12-AIHP487

Primary: 60F99 , 60G50

Keywords: Area of excursion , Area of random walk , Integrated random walk , One-sided exit probability , Persistence , Sparre–Andersen theorem , Stable excursion , Unilateral small deviations

Rights: Copyright © 2014 Institut Henri Poincaré

Vol.50 • No. 1 • February 2014
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