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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Limit theorems for convolution iterates of a probability measure on completely simple or compact semigroups
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by A. Mukherjea PDF
Trans. Amer. Math. Soc. 225 (1977), 355-370 Request permission

Abstract:

This paper extends the study (initiated by M. Rosenblatt) of the asymptotic behavior of the convolution sequence of a probability measure on compact or completely simple semigroups. Let S be a locally compact second countable Hausdorff topological semigroup. Let $\mu$ be a regular probability measure on the Borel subsets of S such that S does not have a proper closed subsemigroup containing the support F of $\mu$. It is shown in this paper that when S is completely simple with its usual product representation $X \times G \times Y$, then the convolution sequence ${\mu ^n}$ converges to zero vaguely if and only if the group factor G is noncompact. When the group factor G is compact, ${\mu ^n}$ converges weakly if and only if ${\underline {\lim } _{n \to \infty }}{F^n}$ is nonempty. This last result remains true for an arbitrary compact semigroup S generated by F. Furthermore, we show that in this case there exist elements ${a_n} \in S$ such that ${\mu ^n} \ast {\delta _{{a_n}}}$ converges weakly, where ${\delta _{{a_n}}}$ is the point mass at ${a_n}$. This result cannot be extended to the locally compact case, even when S is a group.
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 225 (1977), 355-370
  • MSC: Primary 60B15
  • DOI: https://doi.org/10.1090/S0002-9947-1977-0423458-2
  • MathSciNet review: 0423458