1 April 2020 Deviation inequalities for random walks
P. Mathieu, A. Sisto
Duke Math. J. 169(5): 961-1036 (1 April 2020). DOI: 10.1215/00127094-2019-0067


We study random walks on groups, with the feature that, roughly speaking, successive positions of the walk tend to be “aligned.” We formalize and quantify this property by means of the notion of deviation inequalities. We show that deviation inequalities have several consequences, including central limit theorems, the local Lipschitz continuity of the rate of escape and entropy, as well as linear upper and lower bounds on the variance of the distance of the position of the walk from its initial point. In the second part of this article, we show that the (exponential) deviation inequality holds for measures with exponential tail on acylindrically hyperbolic groups. These include nonelementary (relatively) hyperbolic groups, mapping class groups, many groups acting on CAT(0) spaces, and small cancellation groups.


Download Citation

P. Mathieu. A. Sisto. "Deviation inequalities for random walks." Duke Math. J. 169 (5) 961 - 1036, 1 April 2020. https://doi.org/10.1215/00127094-2019-0067


Received: 21 November 2017; Revised: 3 July 2019; Published: 1 April 2020
First available in Project Euclid: 14 March 2020

zbMATH: 07198469
MathSciNet: MR4079419
Digital Object Identifier: 10.1215/00127094-2019-0067

Primary: 60G50
Secondary: 20F65 , 20F67

Keywords: Entropy , Girsanov theorem , hyperbolic groups , Random walks , Rate of escape

Rights: Copyright © 2020 Duke University Press


This article is only available to subscribers.
It is not available for individual sale.

Vol.169 • No. 5 • 1 April 2020
Back to Top