Let $G$ be a group and let $A:=C^*(G)$  be the C*-algebra of the group $G$. This is a C*-algebra whose elements are complex-valued functions on the group $G$. We define operations on $A$ in the ordinary way save for multiplication

$\displaystyle (fg)(s)=\sum_{t\in G}f(t)g(t^{-1}s)$,

and the adjoint $f^*(s)=\overline{f(s^{-1})}$. Note that the above multiplication is the same as defining $\delta_s\delta_t=\delta_{st}$ and extending via linearity. Thence $A$ is abelian if and only if $G$ is.

To give the structure of a quantum group we define the following linear maps:

$\Delta:A\rightarrow A\otimes A$$\Delta(\delta_s)=\delta_s\otimes\delta_s$.

$\displaystyle \varepsilon:A\rightarrow \mathbb{C}$$\varepsilon(\delta_s)=1.$

$S:A\rightarrow A$$S(\delta_s)=\delta_{s^{-1}}$.

The functional $h:A\rightarrow \mathbb{C}$ defined by $h=\mathbf{1}_{\{\delta_e\}}$ is the Haar state on $A$. It is very easy to write down the $j_n$:

$\displaystyle j_n(\delta_s)=\bigotimes_{i=0}^n\delta_s$.

To do probability theory consider states $\varepsilon,\,\phi$ on $A$ and form the product state:

$\displaystyle \Psi=\varepsilon\otimes\bigotimes_{i=1}^\infty \phi$.

Whenever $\phi$ is a state of $A$ such that $\phi(\delta_s)=1$ implies that $s=e$, then the distribution of the random variables $j_n$ converges to $h$.

At the moment we will use the one-norm to measure the distance to stationary:

$d(\Psi\circ j_n,h)=\|\Psi\circ j_n-h\|_1$.

A quick calculation shows that:

$d(\Psi\circ j_n,h)=\sum_{s\neq e}|\phi(\delta_s)|^n$.

When, for example, $\phi(\delta_s)=2/m^2$ when $s$ are transpositions in $S_m$, then we have

$d(\Psi\circ j_n,h)={m\choose 2}\left(\frac{2}{m^2}\right)^n$.