Random Walks on Reductive Groups

Provides a self-contained introduction to the products of independent identically distributed random matrices and to their Lyapunov exponents

Explains the relevance of the theory of reductive algebraic groups and the theory of bounded operators in Banach spaces to the study of random matrices

Contains a proof of the Local Limit Theorem for the norm of the products of independent identically distributed random matrices

Inhalt
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.