Convergence of $\frac1m(I+A+A^2+\cdots+A^{m-1})$ Let $A$ be an $n\times n$ matrix of nonnegative entries such that $A_{i1}+A_{i2}+\cdots+A_{in}=1$ for all $i\in\{1,2,\ldots,n\}$. What does $A$ have to satisfy so that the sequence
$$M_m=\frac1m(I+A+A^2+\cdots+A^{m-1})$$
converge?
I believe this converges for almost all $A$. It should converge to a matrix $M$ such that $MA=M$ and $M_{i1}+M_{i2}+\cdots+M_{in}=1$ for all $i\in\{1,2,\ldots,n\}$. I am pretty sure all of the rows of $M$ are equal most of the time, but I haven't found a rigorous proof. For what I need, I'm interested in something like $|M_m-M|<\epsilon\ll 1$ (for some norm $|\cdot|$, taxicab norm preferred). Thanks in advance!

Your matrix is stochastic, if it is also primitive then the largest eigenvalue is $1$, and all the other eigenvalues satisfy 
$$ | \lambda | <1 \,.$$
What you need is $\#5$ from here:
Properties of Perron-Frobenius Eigenvalue
What is says is the following:
Let $A$ be any primitive matrix and $r$ be the Perron-Frobenius eigenvalue (in your case $r=1$).
Then 
$$\frac{1}{m} \sum_{i=0}^m \frac{A^i}{r^i} \to vw^t$$
where $v,w$ are the left and right PF-eigenvectors normalized by $w^t v=1$.
There is a reference to this result....
P.S. In order to be Primitive, all entries need to be positive. If this doesn't happen, just check the section "Perron projection as a limit: $A^k/r^k$", it actually have the idea of the proof of this result under pretty general settings....
