Probability Measures on Semigroups

by Hognas, Goran; Mukherjea, Arunava

• ISBN13: 9780387775470
• ISBN10: 0387775471
• Edition: 2nd
• Format: Hardcover
• Copyright: 2010-11-01
• Publisher: Springer Verlag

Summary

This second edition presents up-to-date material on the theory of weak convergance of convolution products of probability measures in semigroups, the theory of random walks on semigroups, and their applications to products of random matrices. In addition, this unique work examines the essentials of abstract semigroup theory and its application to concrete semigroups of matrices. This substantially revised text includes exercises at various levels at the end of each section and includes the best available proofs on the most important theorems used in a book, making it suitable for a one semester course on semigroups. In addition, it could also be used as a main text or supplementary material for courses focusing on probability on algebraic structures or weak convergence. This book is ideally suited to graduate students in mathematics, and students in other fields, such as engineering and the sciences with an interest in probability. Students in statistics using advanced probability will also find this book useful.

Table of Contents

Semigroups	p. 1
Introduction	p. 1
Homomorphisms, Quotients, and Products	p. 6
Semigroups with Zero	p. 10
The Rees-Suschkewitsch Representation Theorem	p. 12
Topological Semigroups	p. 22
Semigroups of Matrices	p. 33
Semigroups of Infinite Dimensional Matrices	p. 53
Notes and Comments	p. 60
Probability Measures on Topological Semigroups	p. 63
Introduction	p. 63
Invariant and Idempotent Probability Measures	p. 64
Weak Convergence of Convolution Products of Probability Measures	p. 83
Weak Convergence of Convolution Products of Nonidentical Probability Measures	p. 137
Notes and Comments	p. 168
Random Walks on Semigroups	p. 171
Introduction	p. 171
Discrete Semigroups	p. 180
Locally Compact Groups	p. 198
Compact Semigroups	p. 219
Completely Simple Semigroups	p. 241
Notes and Comments	p. 247
Random Matrices	p. 253
Introduction	p. 253
Recurrent Random Walks in Nonnegative Matrices	p. 254
Tightness of Products of I.I.D. Random Matrices: Weak Convergence	p. 284
Invariant Measures for Random Walks in Nonnegative Matrices: Laws of Large Numbers	p. 333
Notes and Comments	p. 355
Products of I.I.D. Random Stochastic Matrices: Their Skeletons and Convergence in Distribution	p. 359
An Example Due to Chamayou and Letac	p. 365
Asymptotic Behavior of XnXn-1 ... X0u for I.I.D. Random Nonnegative Matrices	p. 369
Continuous Singularity of the Limit Distribution of Products of I.I.D. Stochastic Matrices	p. 383
Rate of Decay of Concentration Functions on Discrete Groups	p. 397
Preliminaries	p. 397
Irreducible Measures	p. 404
Adapted Measures	p. 408
References	p. 413
Index	p. 423
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