Probability Measures on Semigroups: Convolution Products, Random Walks and Random Matrices

Beschreibung

A Scientific American article on chaos, see Crutchfield et al. (1986), illus trates a very persuasive example of recurrence. A painting of Henri Poincare, or rather a digitized version of it, is stretched and cut to produce a mildly distorted image of Poincare. The same procedure is applied to the distorted image and the process is repeated over and over again on the successively more and more blurred images. After a dozen repetitions nothing seems to be left of the original portrait. Miraculously, structured images appear briefly as we continue to apply the distortion procedure to successive images. After 241 iterations the original picture reappears, unchanged! Apparently the pixels of the Poincare portrait were moving about in accor dance with a strictly deterministic rule. More importantly, the set of all pixels, the whole portrait, was transformed by the distortion mechanism. In this exam ple the transformation seems to have been a reversible one since the original was faithfully recreated. It is not very farfetched to introduce a certain amount of randomness and irreversibility in the above example. Think of a random miscoloring of some pixels or of inadvertently giving a pixel the color of its neighbor. The methods in this book are geared towards being applicable to the asymp totics of such transformation processes. The transformations form a semigroup in a natural way; we want to investigate the long-term behavior of random elements of this semigroup.
This second edition presents up-to-date material on the theory of weak convergence of convolution products of probability measures on semigroups, the theory of random walks on semigroups, and their applications to products of random matrices. This substantially revised text includes exercises at various levels at the end of each section and includes the best available proofs of the most important theorems used in the book, making it suitable for a one semester course on semigroups.

This second edition presents up-to-date material on the theory of weak convergence of convolution products of probability measures on 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 of the most important theorems used in the 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 advance probability will also find this book useful.

Autor*in

Göran Högnäs

Themen in »Probability Measures on Semigroups: Convolution Products, Random Walks and Random Matrices«

applied mathematics group theory mathematics probability

Stimmen zu »Probability Measures on Semigroups: Convolution Products, Random Walks and Random Matrices«

`A well-written book...This is elegant mathematics, motivated by examples and presented in an accessible way that engages the reader.'
International Statistics Institute, December 1996

`This beautiful book...guides the reader through the most important developments...a valuable addition to the library of the probabilist, and a must for anybody interested in probability on algebraic structures.'
Zentralblatt für Mathematik und ihre Grenzgebiete-Mathematical Abstracts

`This well-written volume, by two of the most successful workers in the field....deserves to become the standard introduction for beginning researchers in this field.'
Journal Royal Statistical Society

()

Details

Link teilen