fundamental theorem of demography

Let $A_t$ be a sequence of $n\times n$ nonnegative primitive matrices. Suppose that $A_t\to A_{\mathrm{\infty }}$, with $A_{\mathrm{\infty }}$ also a nonnegative primitive matrix. Define the sequence $x_{t+1}=A_t{x}_t$, with $x_t\in {ℝ}^n$.If $x_0\ge 0$, then

$$\underset{t\to \mathrm{\infty }}{lim}\frac{x_t}{\parallel x_t\parallel }=p$$

where $p$ is the normalized ($\parallel p\parallel =1$) eigenvector associated to the eigenvalue of $A_{\mathrm{\infty }}$ (also called the Perron-Frobenius eigenvector of $A_{\mathrm{\infty }}$).