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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).

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maybe you want the perron frobenius thm, en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem –  mike Aug 7 '13 at 17:02
    
No I looked into it but it doesn't seem to solve the problem (unless I missed something). –  Denis Aug 7 '13 at 17:03
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