Given a compact Hausdorff topological group $G$ and probability measures $\mu$ and $\tau$ on the Borel sets of $G$, their convolution is the probability measure


$(\tau\*\mu)(A)=\int\int1\_A(xy)d\tau(x)d\mu(y)$.


A useful interpretation of this is that $G$ is a permutation group of some deck of cards, $\mu$ describes the distribution of arrangements the deck could currently be in, $\tau$ describes a shuffling scheme i.e. the distribution of possible shuffles a dealer may apply to randomize the deck. The convolution $\tau\*\mu$ is then the distribution of the arrangements of the deck after the dealer has shuffled it. For example if $\tau$ and $\mu$ are Dirac measures supported at $g\_1$ and $g\_2$ respectively then $\tau\*\mu$ is the Dirac measure supported at $g\_1g\_2$.


It is helpful to note that if $\tau\_0$ is the unique Haar probability measure on $G$ and $\mu$ is another probability measure, then $\tau\_0\*\mu=\tau\_0$. The interpretation here is that $\tau\_0$ is the best shuffling scheme: every arrangement of the deck is equally likely after applying it, independent of how the deck is initially arranged. Obviously it's desirable for a shuffler to use such a scheme, or at least a scheme which converges to $\tau\_0$. This is my motivation.


I am interested in the following questions: 


1. For which distributions $\tau$ does the sequence of convolutions $\tau^{\*n}=\tau\*\tau\*\ldots\*\tau$ converge, let's say weakly, to the Haar distribution $\tau\_0$? My first guess is these are at least the distributions whose support generates $G$. Related to this, can the speed of convergence be measured? I know of such a result for the usual central limit theorem due to Berry and Esseen.
2. Which distributions $\mu$ on $G$ are infinitely divisible in the sense that for any $n\geq 1$ there is a distribution $\tau\_n$ such that $\mu=\tau\_n^{\*n}$? For example, the Haar distribution is infinitely divisible, but there are more: if $g\in G$ always has an $n$-th root $g\_n$, then the Dirac measure supported at $g$ is infinitely divisible: $\delta\_g = \delta\_{g\_n}^{\*n}$. What are all the infinitely divisible distributions for $G=S^1$?


If someone can point me toward a reference for any of this I would much appreciate it, I don't know much harmonic analysis.



Central limit type theorems for compact Hausdorff topological groups?
For instance, in the recent [paper by Harremoes, page 12](https://arxiv.org/abs/0901.0015) one finds this: 



> 
> **Corollary 20.** Let $P$ be a probability measure on the compact group $G$ with Haar probability measure
>  $U$. Assume that the support of $P$ is not contained in any coset of a proper subgroup of $G$. Then $P^{\*n}$ 
>  converges to $U$ in the weak topology. 
> 
> 
> 


(See also the note in first paragraph on page 5 of that paper.) 


Some results on a geometric rate of the convergence are given in Section 6 of the same paper. 


It also contains a number of useful references. 


