Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Positivity of integrated random walks

Vladislav Vysotsky

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Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 50, Number 1 (2014), 195-213.

Dates
First available in Project Euclid: 1 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1388545271

Digital Object Identifier
doi:10.1214/12-AIHP487

Mathematical Reviews number (MathSciNet)
MR3161528

Zentralblatt MATH identifier
1293.60053

Subjects
Primary: 60G50: Sums of independent random variables; random walks 60F99: None of the above, but in this section

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

Citation

Vysotsky, Vladislav. Positivity of integrated random walks. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 1, 195--213. doi:10.1214/12-AIHP487. https://projecteuclid.org/euclid.aihp/1388545271.


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