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

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 original work presents up-to-date information on three major topics in mathematics research: the theory of weak convergence of convolution products of probability measures in semigroups; the theory of random walks with values in semigroups; and the applications of these theories to products of random matrices. The authors introduce the main topics through the fundamentals of abstract semigroup theory and significant research results concerning its application to concrete semigroups of matrices. The material is suitable for a two-semester graduate course on weak convergence and random walks. It is assumed that the student will have a background in Probability Theory, Measure Theory, and Group Theory. The authors present up-to-date information 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. Graduate students as well as researchers in pure and applied mathematics will appreciate this well-illustrated work. Front Matter....Pages i-xii Semigroups....Pages 1-65 Probability Measures on Topological Semigroups....Pages 67-172 Random Walks on Semigroups....Pages 173-262 Random Matrices....Pages 263-382 Back Matter....Pages 383-388