I have this little problem that has been nagging me for a couple of months now. It occurred to me when considering the fairness of card shuffling methods. Here's my best attempt at formalizing it:
Let $p$ be a probability measure on the symmetric group $S_n$. Consider the composition:
$$\Sigma_N = \sigma_1 \cdot \ldots \cdot \sigma_N $$
Where the $\sigma_i$ are all independent random permutations of $S_n $ with distribution $p $. The question is:
For which possible $p$s does the distribution of $\Sigma_N $ converge for $N\rightarrow\infty $ to the uniform distribution on $S_n $?
As an example, for $n=2$ convergence happens for all $p $ except the two Kronecker deltas.
My progress on this is basically nil. Any suggestions?