Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben

2016 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Random Walks on Reductive Groups

verfasst von: Yves Benoist, Jean-François Quint

Verlag: Springer International Publishing

Buchreihe : Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten
insite
SUCHEN

Über dieses Buch

The classical theory of Random Walks describes the asymptotic behavior of sums of independent identically distributed random real variables. This book explains the generalization of this theory to products of independent identically distributed random matrices with real coefficients.
Under the assumption that the action of the matrices is semisimple – or, equivalently, that the Zariski closure of the group generated by these matrices is reductive - and under suitable moment assumptions, it is shown that the norm of the products of such random matrices satisfies a number of classical probabilistic laws.
This book includes necessary background on the theory of reductive algebraic groups, probability theory and operator theory, thereby providing a modern introduction to the topic.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
The introduction is a dialog between the reader and the authors explaining the content of the book in a light style and in an easily understandable language.
Yves Benoist, Jean-François Quint

The Law of Large Numbers

Frontmatter
Chapter 2. Stationary Measures
Abstract
We recall basic facts on Markov chains, their invariant probability measures and their forward dynamical system. We also recall the Markov chains associated to a random walk and their backward dynamical system.
Yves Benoist, Jean-François Quint
Chapter 3. The Law of Large Numbers
Abstract
We prove the Law of Large Numbers for a real valued cocycle over a semigroup action.
Yves Benoist, Jean-François Quint
Chapter 4. Linear Random Walks
Abstract
We study random walks on the linear groups. We prove the Law of Large Numbers for the norm of matrices and for the norm of vectors. We also prove the positivity of the first Lyapunov exponent.
Yves Benoist, Jean-François Quint
Chapter 5. Finite Index Subsemigroups
Abstract
We relate random walks on groups and the induced random walks on finite index subgroups.
Yves Benoist, Jean-François Quint

Reductive Groups

Frontmatter
Chapter 6. Loxodromic Elements
Abstract
We prove the existence of loxodromic elements in Zariski dense subgroups of semisimple real Lie groups.
Yves Benoist, Jean-François Quint
Chapter 7. The Jordan Projection of Semigroups
Abstract
For Zariski dense subgroups of semisimple real Lie groups, we prove the convexity and non-degeneracy of the limit cone. We also prove the density of the group spanned by the Jordan projections.
Yves Benoist, Jean-François Quint
Chapter 8. Reductive Groups and Their Representations
Abstract
We recall a few basic facts on reductive groups over local fields, their algebraic representations, their flag varieties, their Cartan projection and their Iwasawa cocycle.
Yves Benoist, Jean-François Quint
Chapter 9. Zariski Dense Subsemigroups
Abstract
We study Zariski dense subgroups in products of algebraic reductive groups over local fields.
Yves Benoist, Jean-François Quint
Chapter 10. Random Walks on Reductive Groups
Abstract
We apply the previous results to random walks in products of algebraic reductive groups over local fields. We prove the Law of Large Numbers for both the Iwasawa cocycle and the Cartan projection. We prove also the regularity of the Lyapunov vector.
Yves Benoist, Jean-François Quint

The Central Limit Theorem

Frontmatter
Chapter 11. Transfer Operators over Contracting Actions
Abstract
We focus on cocycles over a contracting action. We study the spectral properties of the corresponding complex transfer operators for small values of the parameter.
Yves Benoist, Jean-François Quint
Chapter 12. Limit Laws for Cocycles
Abstract
We prove the Central Limit Theorem, the Law of Iterated Logarithm and the Large Deviation Principle for a cocycle over a contracting action.
Yves Benoist, Jean-François Quint
Chapter 13. Limit Laws for Products of Random Matrices
Abstract
We apply the previous results to random walks in products of algebraic reductive groups over local fields. We prove the Central Limit Theorem, the Law of Iterated Logarithm and the Large Deviation Principle for the Iwasawa cocycle and for the Cartan projection.
Yves Benoist, Jean-François Quint
Chapter 14. Regularity of the Stationary Measure
Abstract
We give a short proof of the Hölder regularity of the stationary measure on the flag variety. We apply it to the random walks on the linear groups. We prove the Law of Large Numbers for the coefficients and for the spectral radius. We also prove the Central Limit Theorem, Law of Iterated Logarithm and Large Deviation Principle for the norm of matrices, the norm of vectors, the coefficients and the spectral radius.
Yves Benoist, Jean-François Quint

The Local Limit Theorem

Frontmatter
Chapter 15. The Spectrum of the Complex Transfer Operator
Abstract
We come back to the cocycles over a contracting action. We study more deeply the spectral properties of their complex transfer operators for all imaginary values of the parameter.
Yves Benoist, Jean-François Quint
Chapter 16. The Local Limit Theorem for Cocycles
Abstract
We prove a Local Limit Theorem with moderate deviations for cocycles over a contracting action.
Yves Benoist, Jean-François Quint
Chapter 17. The Local Limit Theorem for Products of Random Matrices
Abstract
We apply the previous results to random walks in products of algebraic reductive groups over local fields. We prove a Local Limit Theorem with target and with moderate deviations for the Iwasawa cocycle and for the Cartan projection. For the random walks on the linear groups, we deduce a Local Limit Theorem for the norms of matrices and for the norms of vectors.
Yves Benoist, Jean-François Quint
Backmatter
Metadaten
Titel
Random Walks on Reductive Groups
verfasst von
Yves Benoist
Jean-François Quint
Copyright-Jahr
2016
Electronic ISBN
978-3-319-47721-3
Print ISBN
978-3-319-47719-0
DOI
https://doi.org/10.1007/978-3-319-47721-3