# fundamental theorem of demography

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

 $\lim_{t\to\infty}\frac{x_{t}}{\|x_{t}\|}=p$

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

Title fundamental theorem of demography FundamentalTheoremOfDemography 2013-03-22 13:18:28 2013-03-22 13:18:28 jarino (552) jarino (552) 6 jarino (552) Theorem msc 37A30 msc 92D25 a weak ergodic theorem PerronFrobeniusTheorem