no. 2
Limit laws of transient excited random walks on integers
Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 2, pp. 575-600

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.

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. Auparavant, Kosygina et Zerner [15] ont établi que si la dérive totale moyenne par site, δ, est strictement supérieur 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.

DOI : 10.1214/10-AIHP376
Classification : 60K37, 60F05, 60J80, 60J60
Keywords: excited random walk, limit theorem, stable law, branching process, diffusion approximation 
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     title = {Limit laws of transient excited random walks on integers},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {575--600},
     year = {2011},
     publisher = {Gauthier-Villars},
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     zbl = {1215.60057},
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Kosygina, Elena; Mountford, Thomas. Limit laws of transient excited random walks on integers. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 2, pp. 575-600. doi: 10.1214/10-AIHP376

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