Random Walks on Reductive Groups / by Yves Benoist, Jean-François Quint.

Format:

Book

Author/Creator:

Benoist, Yves., Author.

Quint, Jean-François., Author.

Series:

Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 0071-1136 ; 62

Language:

English

Subjects (All):

Probabilities.

Dynamics.

Ergodic theory.

Topological groups.

Lie groups.

Probability Theory and Stochastic Processes.

Dynamical Systems and Ergodic Theory.

Topological Groups, Lie Groups.

Local Subjects:

Probability Theory and Stochastic Processes.

Dynamical Systems and Ergodic Theory.

Topological Groups, Lie Groups.

Physical Description:

1 online resource (XI, 323 p.)

Edition:

1st ed. 2016.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2016.

Summary:

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.

Contents:

Introduction

Part I The Law of Large Numbers

Stationary measures

The Law of Large Numbers

Linear random walks

Finite index subsemigroups

Part II Reductive groups

Loxodromic elements

The Jordan projection of semigroups

Reductive groups and their representations

Zariski dense subsemigroups

Random walks on reductive groups

Part III The Central Limit Theorem

Transfer operators over contracting actions

Limit laws for cocycles

Limit laws for products of random matrices

Regularity of the stationary measure

Part IV The Local Limit Theorem

The Spectrum of the complex transfer operator

The Local limit theorem for cocycles

The local limit theorem for products of random matrices

Part V Appendix

Convergence of sequences of random variables

The essential spectrum of bounded operators

Bibliographical comments.

Notes:

Includes bibliographical references and index.

ISBN:

3-319-47721-8