Anomalous Behaviour of the Correction to the Central Limit Theorem for a Model of Random Walk in Random Media
Di Persio, L.
Bollettino dell'Unione Matematica Italiana, Tome 3 (2010), p. 179-206 / Harvested from Biblioteca Digitale Italiana di Matematica
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We give a result concerning the correction to the Central Limit Theorem for a Random Walk on the lattice 𝐙2 which interacts with a random environment under a small randomness condition. Our main theorem close a gap which dates back to seminal works by Boldrighini, Minlos and Pellegrinotti, see [3], [8] and [9]. Asymptotic behaviour of the corrections to the average and the covariance matrix in dimension ν=1,2 are also presented.

Publié le : 2010-02-01
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     author = {L. Di Persio},
     title = {Anomalous Behaviour of the Correction to the Central Limit Theorem for a Model of Random Walk in Random Media},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {3},
     year = {2010},
     pages = {179-206},
     zbl = {1202.60164},
     mrnumber = {2605919},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2010_9_3_1_179_0}
}

Di Persio, L. Anomalous Behaviour of the Correction to the Central Limit Theorem for a Model of Random Walk in Random Media. Bollettino dell'Unione Matematica Italiana, Tome 3 (2010) pp. 179-206. http://gdmltest.u-ga.fr/item/BUMI_2010_9_3_1_179_0/

[1] Bérard, J., The almost sure central limit theorem for one-dimensional nearest-neighbour random walks in a space-time random environment, J. Appl. Probab., 41, no. 1 (2004), 83-92. | MR 2036273 | Zbl 1087.60027

[2] Bernabei, M. S., Anomalous behaviour for the random corrections to the cumulants of random walks in fluctuating random media, Probab. Theory Related Fields, 119, no. 3 (2001), 410-432. | MR 1821141 | Zbl 0989.60043

[3] Bernabei, M. S. - Boldrighini, C. - Minlos, R. A. - Pellegrinotti, A., Almost-sure central limit theorem for a model of random walk in fluctuating random environment, Markov Process. Related Fields, 4, no. 3 (1998), 381-393. | MR 1670035 | Zbl 0927.60059

[4] Boldrighini, C. - Cosimi, G. - Frigio, S. - Pellegrinotti, A., Computer simulations for some one-dimensional models of random walks in fluctuating random environment, J. Stat. Phys., 121, no. 3-4 (2005), 361-372. | MR 2213461 | Zbl 1149.82321

[5] Boldrighini, C. - Kondratiev, Yu. G. - Minlos, R. A. - Pellegrinotti, A. - Zhizhina, E. A., Random jumps in evolving random environment, Markov Process. Related Fields, 14, no. 4 (2008), 543-570. | MR 2473767 | Zbl 1160.60027

[6] Boldrighini, C. - Minlos, R. A. - Nardi, F. R. - Pellegrinotti, A., Asymptotic decay of correlations for a random walk in interaction with a Markov field, Mosc. Math. J., 5, no. 3 (2005), 507-522, 742. | MR 2241810 | Zbl 1116.60058

[7] Boldrighini, C. - Minlos, R. A. - Nardi, F. R. - Pellegrinotti, A., Asymptotic decay of correlations for a random walk on the lattice 𝐙d in interaction with a Markov field, Mosc. Math. J., 8, no. 3 (2008), 419-431, 615. | MR 2483218 | Zbl 1155.60045

[8] Boldrighini, C. - Minlos, R. A. - Pellegrinotti, A., Almost-sure central limit theorem for a Markov model of random walk in dynamical random environment, Probab. Theory Related Fields, 109, no. 2 (1997), 245-273. | MR 1477651 | Zbl 0888.60061

[9] Boldrighini, C. - Minlos, R. A. - Pellegrinotti, A., Central limit theorem for a random walk in dynamical environment: integral and local, Proceedings of the Third Ukrainian-Scandinavian Conference in Probability Theory and Mathematical Statistics (Kiev, 1999), vol. 5 (1999), 16-28. | MR 2018394 | Zbl 0994.60056

[10] Boldrighini, C. - Minlos, R. A. - Pellegrinotti, A., Random walks in quenched i.i.d. space-time random environment are always a.s. diffusive, Probab. Theory Related Fields, 129, no. 1 (2004), 133-156. | MR 2052866 | Zbl 1062.60044

[11] Boldrighini, C. - Minlos, R. A. - Pellegrinotti, A., Random walks in a random (fluctuating) environment, Uspekhi Mat. Nauk, 62, no. 4 (2007), 27-76. | MR 2358736 | Zbl 1145.60052

[12] Boldrighini, C. - Pellegrinotti, A., T-1/4-noise for random walks in dynamic environment on 𝐙, Mosc. Math. J., 1, no. 3 (2001), 365-380, 470-471. | MR 1877598 | Zbl 1006.60100

[13] Brémont, J., On some random walks on 𝐙 in random medium, Ann. Probab., 30, no. 3 (2002), 1266-1312. | MR 1920108

[14] Brémont, J., Random walks in random medium on 𝐙 and Lyapunov spectrum, Ann. Inst. H. Poincaré Probab. Statist., 40, no. 3 (2004), 309-336. | MR 2060456

[15] Brémont, J., One-dimensional finite range random walk in random medium and invariant measure equation, Ann. Inst. Henri Poincaré Probab. Stat., 45, no. 1 (2009), 70-103. | MR 2500229

[16] Chow, Y. S. - Teicher, H., Probability theory, third ed., Springer Texts in Statistics (Springer-Verlag, New York, 1997), Independence, interchangeability, martingales. | MR 1476912

[17] Dolgopyat, D. - Keller, G. - Liverani, C., Random walk in Markovian environment, Ann. Probab., 36, no. 5 (2008), 1676-1710. | MR 2440920 | Zbl 1192.60110

[18] Gīhman, Ĭ. Ī. - Skorohod, A. V., The theory of stochastic processes. I, English ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 210 (Springer-Verlag, Berlin, 1980), Translated from the Russian by Samuel Kotz. | MR 636254 | Zbl 0531.60001

[19] Ibragimov, I. A. - Linnik, Yu. V., Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971, With a supplementary chapter by I. A. Ibragimov and V. V. Petrov, Translation from the Russian edited by J. F. C. Kingman. | MR 322926 | Zbl 0219.60027

[20] Zeitouni, O., Random walks in random environments, J. Phys. A, 39, no. 40 (2006), R433-R464. | MR 2261885 | Zbl 1108.60085