Limit laws of transient excited random walks on integers

Elena Kosygina; Thomas Mountford

Annales de l'I.H.P. Probabilités et statistiques (2011)

  • Volume: 47, Issue: 2, page 575-600
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Kosygina, Elena, and Mountford, Thomas. "Limit laws of transient excited random walks on integers." Annales de l'I.H.P. Probabilités et statistiques 47.2 (2011): 575-600. <http://eudml.org/doc/239308>.

@article{Kosygina2011,
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&gt;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.},
author = {Kosygina, Elena, Mountford, Thomas},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {excited random walk; limit theorem; stable law; branching process; diffusion approximation},
language = {eng},
number = {2},
pages = {575-600},
publisher = {Gauthier-Villars},
title = {Limit laws of transient excited random walks on integers},
url = {http://eudml.org/doc/239308},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Kosygina, Elena
AU - Mountford, Thomas
TI - Limit laws of transient excited random walks on integers
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2011
PB - Gauthier-Villars
VL - 47
IS - 2
SP - 575
EP - 600
AB - 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&gt;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.
LA - eng
KW - excited random walk; limit theorem; stable law; branching process; diffusion approximation
UR - http://eudml.org/doc/239308
ER -

References

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