Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Exit times for integrated random walks

Denis Denisov and Vitali Wachtel

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Abstract

We consider a centered random walk with finite variance and investigate the asymptotic behaviour of the probability that the area under this walk remains positive up to a large time $n$. Assuming that the moment of order $2+\delta$ is finite, we show that the exact asymptotics for this probability is $n^{-1/4}$. To show this asymptotics we develop a discrete potential theory for integrated random walks.

Résumé

Nous considérons une marche aléatoire centrée de variance finie et étudions le comportement asymptotique de la probabilité que l’aire sous la marche reste positive jusqu’à un grand temps $n$. Si le moment d’ordre $2+\delta$ est fini, nous montrons que cette probabilité décroit comme $n^{-1/4}$. Pour prouver ce comportement asymptotique, nous développons une théorie du potentiel discrète pour des marches aléatoires intégrées.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 51, Number 1 (2015), 167-193.

Dates
First available in Project Euclid: 14 January 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1421244402

Digital Object Identifier
doi:10.1214/13-AIHP577

Mathematical Reviews number (MathSciNet)
MR3300967

Zentralblatt MATH identifier
1310.60049

Subjects
Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60F17: Functional limit theorems; invariance principles

Keywords
Markov chain Exit time Harmonic function Weyl chamber Normal approximation Kolmogorov diffusion

Citation

Denisov, Denis; Wachtel, Vitali. Exit times for integrated random walks. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 1, 167--193. doi:10.1214/13-AIHP577. https://projecteuclid.org/euclid.aihp/1421244402


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