Limit laws of transient excited random walks on integers
Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 2, p. 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 : https://doi.org/10.1214/10-AIHP376
Classification:  60K37,  60F05,  60J80,  60J60
Keywords: excited random walk, limit theorem, stable law, branching process, diffusion approximation
@article{AIHPB_2011__47_2_575_0,
     author = {Kosygina, Elena and Mountford, Thomas},
     title = {Limit laws of transient excited random walks on integers},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {2},
     year = {2011},
     pages = {575-600},
     doi = {10.1214/10-AIHP376},
     zbl = {1215.60057},
     mrnumber = {2814424},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2011__47_2_575_0}
}
Kosygina, Elena; Mountford, Thomas. Limit laws of transient excited random walks on integers. Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 2, pp. 575-600. doi : 10.1214/10-AIHP376. http://www.numdam.org/item/AIHPB_2011__47_2_575_0/

[1] A.-L. Basdevant and A. Singh. On the speed of a cookie random walk. Probab. Theory Related Fields 141 (2008) 625-645. | MR 2391167 | Zbl 1141.60383

[2] A.-L. Basdevant and A. Singh. Rate of growth of a transient cookie random walk. Electron. J. Probab. 13 (2008) 811-851. | MR 2399297 | Zbl 1191.60107

[3] I. Benjamini and D. B. Wilson. Excited random walk. Electron. Comm. Probab. 8 (2003) 86-92. | MR 1987097 | Zbl 1060.60043

[4] J. Bérard and A. Ramírez. Central limit theorem for the excited random walk in dimension d≥2. Elect. Comm. in Probab. 12 (2007) 303-314. | MR 2342709 | Zbl 1128.60082

[5] L. Chaumont and R. A. Doney. Pathwise uniqueness for perturbed versions of Brownian motion and reflected Brownian motion. Probab. Theory Related Fields 113 (1999) 519-534. | MR 1717529 | Zbl 0945.60082

[6] B. Davis. Brownian motion and random walk perturbed at extrema. Probab. Theory Related Fields 113 (1999) 501-518. | MR 1717528 | Zbl 0930.60041

[7] D. Dolgopyat. Central limit theorem for excited random walk in the recurrent regime. Preprint, 2008. | MR 2831235

[8] R. Durrett. Probability: Theory and Examples, 2nd edition. Duxbury Press, Belmont, CA, 1996. | MR 1609153 | Zbl 1202.60002

[9] A. Gut. Stopped Random Walks. Limit Theorems and Applications. Applied Probability. A Series of the Applied Probability Trust 5. Springer, New York, 1988. | MR 916870 | Zbl 0634.60061

[10] S. Ethier and T. Kurtz. Markov Processes. Wiley, New York, 1986. | MR 838085 | Zbl 0592.60049

[11] W. Feller. An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edition. Wiley, New York, 1971. | MR 270403 | Zbl 0138.10207

[12] S. K. Formanov, M. T. Yasin and S. V. Kaverin. Life spans of Galton-Watson processes with migration (Russian). In Asymptotic Problems in Probability Theory and Mathematical Statistics 117-135. T. A. Azlarov and S. K. Formanov (Eds). Fan, Tashkent, 1990. | MR 1142599

[13] R. Van Der Hofstad and M. Holmes. Monotonicity for excited random walk in high dimensions, 2008. Available at arXiv:0803.1881v2 [math.PR]. | MR 2594356 | Zbl 1193.60123

[14] H. Kesten, M. V. Kozlov and F. Spitzer. A limit law for random walk in random environment. Compos. Math. 30 (1975) 145-168. | Numdam | MR 380998 | Zbl 0388.60069

[15] E. Kosygina and M. P. W. Zerner. Positively and negatively excited random walks, with branching processes. Electron. J. Probab. 13 (2008) 1952-1979. | MR 2453552 | Zbl 1191.60113

[16] M. V. Kozlov. Random walk in a one-dimensional random medium. Teor. Verojatn. Primen. 18 (1973) 406-408. | MR 319274 | Zbl 0299.60054

[17] G. Kozma. Excited random walk in three dimensions has positive speed. Preprint, 2003. Available at arXiv:0310305v1 [math.PR].

[18] T. Mountford, L. P. R. Pimentel and G. Valle. On the speed of the one-dimensional excited random walk in the transient regime. ALEA 2 (2006) 279-296. | MR 2285733 | Zbl 1115.60103

[19] M. P. W. Zerner. Multi-excited random walks on integers. Probab. Theory Related Fields 133 (2005) 98-122. | MR 2197139 | Zbl 1076.60088

[20] M. P. W. Zerner. Recurrence and transience of excited random walks on ℤd and strips. Electron. Comm. Probab. 11 (2006) 118-128. | MR 2231739 | Zbl 1112.60086