Let $A$ be a square matrix with nonnegative integer coefficients.
Is there a simple way to prove that there is a "period" $d$ such that for all $0\leq r<d$, the coefficient $a_{i,j,n}$ at position $(i,j)$ in the matrix $A^{nd+r}$ is asymptotically (with respect to $n$) equivalent to $x n^y z^n$, where $x,y,z$ are nonnegative constants depending on $i,j,d,r$.
The period is needed to avoid oscillating behaviours like $(-1)^n$, or cycles of bigger length.
I found this paper which should help, but I'm having trouble formalizing a proof: http://www.ams.org/journals/jams/2000-13-04/S0894-0347-00-00342-8/S0894-0347-00-00342-8.pdf).