Random Walks on Reductive Groups
Springer International Publishing (Verlag)
978-3-319-47719-0 (ISBN)
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.
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.
"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)
| Erscheinungsdatum | 12.11.2016 |
|---|---|
| Reihe/Serie | Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics |
| Zusatzinfo | XI, 323 p. |
| Verlagsort | Cham |
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika |
| Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik | |
| Schlagworte | algebraic group • central limit theorem • Dynamical Systems and Ergodic Theory • essential spectrum • Groups and group theory • law of large numbers • Local Limit Theorem • Lyapunov exponents • Markov Chain • Martingale • Mathematics • mathematics and statistics • Nonlinear Science • probability and statistics • Probability theory and stochastic processes • Stationary measure • stochastics • Topological Groups, Lie Groups |
| ISBN-10 | 3-319-47719-6 / 3319477196 |
| ISBN-13 | 978-3-319-47719-0 / 9783319477190 |
| Zustand | Neuware |
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