Probability Measures on Topological Semigroups

Abstract

This chapter forms the core of this book. It introduces and develops the key concepts, methods, and results (needed for the rest of this book) that involve convolution products of probability measures and their weak convergence. Most of the results presented here are in their final form. However, there are a number of problems, as the reader will readily discover, which are natural, important, and still waiting to be solved. For example, the asymptotic behavior of the sequence of convolution powers of a probabilitymeasure µ, though reasonably completely known in a compact or discrete semigroup (see Theorems 2.7 and 2.13) is not completely clear in the noncompact nondiscrete situation. Also, the problem of weak convergence of convolution products of a sequence of nonidentical probability measures, even though reasonably resolved in a compact abelian semigroup as well as in a discrete group or a discrete abelian semigroup (see Theorem 2.22 and 2.27, Corollary 2.6) is far from being resolved even in the case of compact groups.