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:

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.

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.