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

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.