## 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