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Probability Measures on Topological Semigroups

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Probability Measures on Semigroups

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

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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 probability measure μ, though reasonably completely known in a compact or discrete semigroup (see Theorems 2.13 and 2.29) 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 2.44, Corollary 2.50 and Theorem 2.51), is far from being resolved even in the case of compact groups.

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Authors and Affiliations

  1. Åbo Akademi University, Åbo, Finland

    Göran Högnäs

  2. University of South Florida, Tampa, Florida, USA

    Arunava Mukherjea

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  1. Göran Högnäs
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  2. Arunava Mukherjea
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Högnäs, G., Mukherjea, A. (1995). Probability Measures on Topological Semigroups. In: Probability Measures on Semigroups. The University Series in Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-2388-5_2

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  • DOI: https://doi.org/10.1007/978-1-4757-2388-5_2

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