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February 2021 Global observables for RW: Law of large numbers
Dmitry Dolgopyat, Marco Lenci, Péter Nándori
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Ann. Inst. H. Poincaré Probab. Statist. 57(1): 94-115 (February 2021). DOI: 10.1214/20-AIHP1072

Abstract

We consider the sums TN=n=1NF(Sn) where Sn is a random walk on Zd and F:ZdR is a global observable, that is, a bounded function which admits an average value when averaged over large cubes. We show that TN always satisfies the weak Law of Large Numbers but the strong law fails in general except for one dimensional walks with drift. Under additional regularity assumptions on F, we obtain the Strong Law of Large Numbers and estimate the rate of convergence. The growth exponents which we obtain turn out to be optimal in two special cases: for quasiperiodic observables and for random walks in random scenery.

Nous considérons la somme TN=n=1NF(Sn), où Sn est une marche aléatoire à valeurs dans Zd et F:ZdR est une observable globale, c’est-à-dire une fonction bornée ayant une valeur moyenne sur de grands cubes. Nous montrons que TN satisfait toujours la loi faible des grands nombres mais la loi forte échoue en général, sauf dans le cas de la marche aléatoire unidimensionnelle avec dérive. Sous certaines hypothèses de régularité supplémentaires, nous obtenons la loi forte des grands nombres et nous estimons la vitesse de convergence. Les exposants que nous obtenons sont optimaux dans deux cas particuliers: pour les observables quasi-périodiques et pour les marches aléatoires en paysage aléatoire.

Acknowledgements

The research of DD was partially sponsored by NSF DMS 1665046. The research of ML was partially supported by PRIN 2017S35EHN (MUR, Italy). The research of PN was partially sponsored by NSF DMS 1800811 and NSF DMS 1952876.

Citation

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Dmitry Dolgopyat. Marco Lenci. Péter Nándori. "Global observables for RW: Law of large numbers." Ann. Inst. H. Poincaré Probab. Statist. 57 (1) 94 - 115, February 2021. https://doi.org/10.1214/20-AIHP1072

Information

Received: 22 February 2019; Revised: 22 March 2020; Accepted: 22 May 2020; Published: February 2021
First available in Project Euclid: 12 March 2021

Digital Object Identifier: 10.1214/20-AIHP1072

Subjects:
Primary: 60F15, 60G50
Secondary: 37A40, 60K37

Rights: Copyright © 2021 Association des Publications de l’Institut Henri Poincaré

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Vol.57 • No. 1 • February 2021
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