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H 1 A brief introduction to concentration inequalities

Auteurs : Chazottes, Jean-René (Auteur de la Conférence)
CIRM (Editeur )

Résumé : $Let (X,T)$ be a dynamical system preserving a probability measure $\mu$. A concentration inequality quantifies how small is the probability for $F(x,Tx,\ldots,T^{n-1}x)$ to deviate from $\int F(x,Tx,\ldots,T^{n-1}x) \mathrm{d}\mu(x)$ by an given amount $u$, where $F:X^n\to\mathbb{R}$ is supposed to be separately Lipschitz. The bound on that probability involves a constant $C$ depending only on the dynamical system (thus independent of $n$), and $\sum_{i=0}^{n-1} \mathrm{Lip}_i(F)^2$. In the best situation, the bound is $\exp(-C u^2/\sum_{i=0}^{n-1} \mathrm{Lip}_i(F)^2)$.
After explaining how to get such a bound for independent random variables, I will show how to prove it for a Gibbs measure on a shift of finite type with a Lipschitz potential, and present examples of functions $F$ to which one can apply the inequality. Finally, I will survey some results obtained for nonuniformly hyperbolic systems modeled by Young towers.

Keywords : transfer operator; non-uniformly hyperbolic dynamical system

Codes MSC :
37A50 - Relations with probability theory and stochastic processes
37D20 - Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37D25 - Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)

 Informations sur la Vidéo Langue : Anglais Date de publication : 21/07/2019 Date de captation : 02/07/2019 Collection : Research schools ; Dynamical Systems and Ordinary Differential Equations ; Probability and Statistics Format : MP4 Durée : 00:59:31 Domaine : Dynamical Systems & ODE ; Probability & Statistics Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2019-07-02_Chazottes.mp4 Informations sur la rencontre Nom de la rencontre : Jean-Morlet Chair 2019 - Research School: Thermodynamic Formalism: Modern Techniques in Smooth Ergodic Theory / Chaire Jean-Morlet 2019 - Ecole : Formalisme thermodynamique : techniques modernes en théorie ergodiqueOrganisateurs de la rencontre : Pollicott, Mark ; Vaienti, SandroDates : 01/07/2019 - 05/07/2019 Année de la rencontre : 2019 URL Congrès : https://www.chairejeanmorlet.com/2019-2-...Citation Data DOI : 10.24350/CIRM.V.19541503 Cite this video as: Chazottes, Jean-René (2019). A brief introduction to concentration inequalities. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19541503 URI : http://dx.doi.org/10.24350/CIRM.V.19541503

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