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

  • Book
  • © 1995

Overview

Authors:
  1. Göran Högnäs

    1. Åbo Akademi University, Åbo, Finland

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    You can also search for this author in PubMed   Google Scholar

  2. Arunava Mukherjea

    1. University of South Florida, Tampa, USA

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    You can also search for this author in PubMed   Google Scholar

  • 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.

Part of the book series: University Series in Mathematics (USMA)

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Table of contents (4 chapters)

  1. Front Matter

    Pages i-xii
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  2. Semigroups

    • Göran Högnäs, Arunava Mukherjea
    Pages 1-65

  3. Probability Measures on Topological Semigroups

    • Göran Högnäs, Arunava Mukherjea
    Pages 67-172

  4. Random Walks on Semigroups

    • Göran Högnäs, Arunava Mukherjea
    Pages 173-262

  5. Random Matrices

    • Göran Högnäs, Arunava Mukherjea
    Pages 263-382

  6. Back Matter

    Pages 383-388
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Keywords

About this book

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.

Reviews

`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

Authors and Affiliations

  • Åbo Akademi University, Åbo, Finland

    Göran Högnäs

  • University of South Florida, Tampa, USA

    Arunava Mukherjea

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