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© 2016

Random Walks on Reductive Groups

Book

Table of contents

  1. Front Matter
    Pages I-XI
  2. Yves Benoist, Jean-François Quint
    Pages 1-16
  3. The Law of Large Numbers

    1. Front Matter
      Pages 17-17
    2. Yves Benoist, Jean-François Quint
      Pages 19-36
    3. Yves Benoist, Jean-François Quint
      Pages 37-49
    4. Yves Benoist, Jean-François Quint
      Pages 51-75
    5. Yves Benoist, Jean-François Quint
      Pages 77-86
  4. Reductive Groups

    1. Front Matter
      Pages 87-87
    2. Yves Benoist, Jean-François Quint
      Pages 89-113
    3. Yves Benoist, Jean-François Quint
      Pages 115-126
    4. Yves Benoist, Jean-François Quint
      Pages 127-145
    5. Yves Benoist, Jean-François Quint
      Pages 147-152
    6. Yves Benoist, Jean-François Quint
      Pages 153-167
  5. The Central Limit Theorem

    1. Front Matter
      Pages 169-169
    2. Yves Benoist, Jean-François Quint
      Pages 171-189
    3. Yves Benoist, Jean-François Quint
      Pages 191-202
    4. Yves Benoist, Jean-François Quint
      Pages 203-222
    5. Yves Benoist, Jean-François Quint
      Pages 223-245
  6. The Local Limit Theorem

    1. Front Matter
      Pages 247-247
    2. Yves Benoist, Jean-François Quint
      Pages 249-258

About this book

Introduction

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.

Keywords

Markov chain Martingale Stationary measure Law of Large Numbers Lyapunov exponents Algebraic group Central Limit Theorem Local Limit Theorem Essential spectrum

Authors and affiliations

  1. 1.Université Paris-Sud OrsayFrance
  2. 2.Institut de Mathématiques de BordeauxUniversité Bordeaux 1 Talence CedexFrance

Bibliographic information

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Reviews

“Benoist and Quint have written an excellent text, one that will surely become a standard reference to introduce students to the fascinating nonabelian extension of the now-classical study of random walks. … I congratulate the authors on their well-written and timely offering, and strongly recommend that libraries order a copy of this excellent text!” (Tushar Das, MAA Reviews, November, 2017)

“This book is an exposition of the tools and perspectives needed to reach the current frontier of research in the field of random matrix products… This is a technical subject, drawing on tools from a diverse range of topics… The authors take time to explain everything at a reasonable pace.” (Radhakrishnan Nair, Mathematical Reviews)


 “This reviewer does not hesitate to consider this book as exceptional.” (Marius Iosifescu, zbMATH, 1366.60002)