# Convergence of iterated stochastic matrices

(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 $0$?

• Is the following analysis helpful? $$||M_n^n-A^n|| = || \sum_{i=0}^{n-1} M_n^{n-i-1} (M_n - A) A^i || \le \sum_{i=0}^{n-1} ||M_n||^{n-i-1} ||M_n - A|| ||A||^i \leq n \alpha^n \to 0$$ I assume that the norm is submultiplicative and the norm of a stochastic matrix is at most 1. In particular, $||M|| = \max_{||x||_1=1} ||xM||_1$ satisfies these properties. – Thomas Mar 29 '14 at 20:52
• If want to use a different norm, you can go via the 1-norm. This seems to answer the question as stated, unless I'm missing something. – Thomas Mar 29 '14 at 23:30
• It does answer the question! I was using the same equality, but missing a norm which is both submultiplicative and where stochastic matrices are unitary. Thank you very much – Nathanaël Fijalkow Mar 30 '14 at 11:39