Known asymptotic behaviour, $\lim_{m \rightarrow \infty} A^m$, of positive and column stochastic matrix $A = [a_{ij}] \in M_n$?

Question

Dear Linear Algebra Experts,

Let us say $A = [a_{ij}] \in M_n$ and $n > 2$, which is a positive matrix $A > 0$ and each element of $A$ is $0 < a_{ij} \leq 1$. Also, $A$ is a column stochastic matrix, i.e., $\sum_i a_{ij} = 1$.

Is there any general or known asymptotic behaviour, $\lim_{m \rightarrow \infty} A^m$, of such a matrix $A$?

If yes, what is it? and does it have a proof?

Thank you very much