(This question has been unsuccessfully asked on mathoverflow: https://mathoverflow.net/questions/161728/convergence-of-iterated-stochastic-matrices)
It is well-known that for a stochastic aperiodic matrix $M$, the sequence of iterated powers $(M^n)_n$ converges.
Here I would like to a have a more precise analysis. Consider now a sequence of stochastic matrices $(M_n)_n$, converging to $A$. We even assume that there exists $0 < \alpha < 1$ such that for all $n$, we have $||M_n - A|| \le \alpha^n$. Is it true that $||M_n^n - A^n||$ converges to $0$?