Probability Measures on Semigroups (eBook)

From the Preface to the First Edition 6
Preface to the Second Edition 10
Contents 12
Chapter 1 Semigroups 14
1.1 Introduction 14
1.2 Homomorphisms, Quotients, and Products 19
1.3 Semigroups with Zero 23
1.4 The Rees–Suschkewitsch Representation Theorem 25
1.5 Topological Semigroups 35
1.6 Semigroups of Matrices 46
1.7 Semigroups of Infinite Dimensional Matrices 66
1.8 Notes and Comments 73
Chapter 2 Probability Measures on Topological Semigroups 76
2.1 Introduction 76
2.2 Invariant and Idempotent Probability Measures 77
2.3 Weak Convergence of Convolution Products of Probability Measures 96
2.4 Weak Convergence of Convolution Products of Nonidentical Probability Measures 150
2.5 Notes and Comments 181
Chapter 3 Random Walks on Semigroups 183
3.1 Introduction 183
3.2 Discrete Semigroups 192
3.3 Locally Compact Groups 210
3.4 Compact Semigroups 231
3.5 Completely Simple Semigroups 253
3.6 Notes and Comments 259
Chapter 4 Random Matrices 264
4.1 Introduction 264
4.2 Recurrent Random Walks in Nonnegative Matrices 265
4.3 Tightness of Products of I.I.D. Random Matrices: Weak Convergence 295
4.4 Invariant Measures for Random Walks in Nonnegative Matrices: Laws of Large Numbers 344
4.5 Notes and Commments 366
Appendix A Products of I.I.D. Random Stochastic Matrices: Their Skeletons and Convergence in Distribution 370
Appendix B An Example Due to Chamayou and Letac 376
Appendix C Asymptotic Behavior of XnXn-1 : : :X0u for I.I.D. Random Nonnegative Matrices 379
Appendix D Continuous Singularity of the Limit Distribution of Products of I.I.D. Stochastic Matrices 392
Appendix E Rate of Decay of Concentration Functions on Discrete Groups, by T.M. Retzlaff 405
E.1 Preliminaries 405
E.2 Irreducible Measures 412
E.3 Adapted Measures 416
References 421
Index 431