Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Limit laws of transient excited random walks on integers

Elena Kosygina and Thomas Mountford

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Abstract

We consider excited random walks (ERWs) on ℤ with a bounded number of i.i.d. cookies per site without the non-negativity assumption on the drifts induced by the cookies. Kosygina and Zerner [15] have shown that when the total expected drift per site, δ, is larger than 1 then ERW is transient to the right and, moreover, for δ>4 under the averaged measure it obeys the Central Limit Theorem. We show that when δ∈(2, 4] the limiting behavior of an appropriately centered and scaled excited random walk under the averaged measure is described by a strictly stable law with parameter δ/2. Our method also extends the results obtained by Basdevant and Singh [2] for δ∈(1, 2] under the non-negativity assumption to the setting which allows both positive and negative cookies.

Résumé

On considère des marches aléatoires excitées sur ℤ avec un nombre borné de cookies i.i.d. à chaque site, ceci sans l’hypothèse de positivité. Auparavent, Kosygina et Zerner [15] ont établi que si la dérive totale moyenne par site, δ, est strictement superieur à 1, alors la marche est transiente (vers la droite) et, de plus, pour δ>4 il y a un théorème central limite pour la position de la marche. Ici, on démontre que pour δ∈(2, 4] cette position, convenablement centrée et réduite, converge vers une loi stable de paramètre δ/2. L’approche permet également d’étendre les résultats de Basdevant et Singh [2] pour δ∈(1, 2] à notre cadre plus général.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 2 (2011), 575-600.

Dates
First available in Project Euclid: 23 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1300887283

Digital Object Identifier
doi:10.1214/10-AIHP376

Mathematical Reviews number (MathSciNet)
MR2814424

Zentralblatt MATH identifier
1215.60057

Subjects
Primary: 60K37: Processes in random environments 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J60: Diffusion processes [See also 58J65]

Keywords
Excited random walk Limit theorem Stable law Branching process Diffusion approximation

Citation

Kosygina, Elena; Mountford, Thomas. Limit laws of transient excited random walks on integers. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 2, 575--600. doi:10.1214/10-AIHP376. https://projecteuclid.org/euclid.aihp/1300887283


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