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Transactions of the American Mathematical Society

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

 

 

Limit theorems for convolution iterates of a probability measure on completely simple or compact semigroups


Author: A. Mukherjea
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
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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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DOI: https://doi.org/10.1090/S0002-9947-1977-0423458-2
Keywords: Locally compact topological group, convolution of measures, weak$ ^\ast$-topology
Article copyright: © Copyright 1977 American Mathematical Society