# Planar Lorentz process in a random scenery

Annales de l'I.H.P. Probabilités et statistiques (2009)

- Volume: 45, Issue: 3, page 818-839
- ISSN: 0246-0203

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topPène, Françoise. "Planar Lorentz process in a random scenery." Annales de l'I.H.P. Probabilités et statistiques 45.3 (2009): 818-839. <http://eudml.org/doc/78046>.

@article{Pène2009,

abstract = {We consider the periodic planar Lorentz process with convex obstacles (and with finite horizon). In this model, a point particle moves freely with elastic reflection at the fixed convex obstacles. The random scenery is given by a sequence of independent, identically distributed, centered random variables with finite and non-null variance. To each obstacle, we associate one of these random variables. We suppose that each time the particle hits an obstacle, it wins the amount given by the random variable associated to the obstacle. We prove a convergence in distribution to a Wiener process for the total amount won by the particle (normalized by $\sqrt\{n\log (n)\}$) when the timen goes to infinity. Such a result has been established by Bolthausen [Ann. Probab.17 (1989) 108–115)] in the case of random walks in ℤ2 given by sums of independent identically distributed random variables. We follow the scheme of his proof. The lack of independence will be compensated by some extensions of the local limit theorem proved by Szász and Varjú in [Ergodic Theory Dynam. Systems24 (2004) 257–278]. This paper answers a question of Szász about the asymptotic behaviour of ∑k=0n−1ζSk where (ζℓ)ℓ is a sequence of i.i.d. centered random variables (with finite and non-null variance) and where Sk is the number of the cell at the kth reflection.},

author = {Pène, Françoise},

journal = {Annales de l'I.H.P. Probabilités et statistiques},

keywords = {Lorentz process; finite horizon; random scenery; limit theorem; billiard; infinite measure},

language = {eng},

number = {3},

pages = {818-839},

publisher = {Gauthier-Villars},

title = {Planar Lorentz process in a random scenery},

url = {http://eudml.org/doc/78046},

volume = {45},

year = {2009},

}

TY - JOUR

AU - Pène, Françoise

TI - Planar Lorentz process in a random scenery

JO - Annales de l'I.H.P. Probabilités et statistiques

PY - 2009

PB - Gauthier-Villars

VL - 45

IS - 3

SP - 818

EP - 839

AB - We consider the periodic planar Lorentz process with convex obstacles (and with finite horizon). In this model, a point particle moves freely with elastic reflection at the fixed convex obstacles. The random scenery is given by a sequence of independent, identically distributed, centered random variables with finite and non-null variance. To each obstacle, we associate one of these random variables. We suppose that each time the particle hits an obstacle, it wins the amount given by the random variable associated to the obstacle. We prove a convergence in distribution to a Wiener process for the total amount won by the particle (normalized by $\sqrt{n\log (n)}$) when the timen goes to infinity. Such a result has been established by Bolthausen [Ann. Probab.17 (1989) 108–115)] in the case of random walks in ℤ2 given by sums of independent identically distributed random variables. We follow the scheme of his proof. The lack of independence will be compensated by some extensions of the local limit theorem proved by Szász and Varjú in [Ergodic Theory Dynam. Systems24 (2004) 257–278]. This paper answers a question of Szász about the asymptotic behaviour of ∑k=0n−1ζSk where (ζℓ)ℓ is a sequence of i.i.d. centered random variables (with finite and non-null variance) and where Sk is the number of the cell at the kth reflection.

LA - eng

KW - Lorentz process; finite horizon; random scenery; limit theorem; billiard; infinite measure

UR - http://eudml.org/doc/78046

ER -

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