fundamental theorem of demographyFundamentalTheoremOfDemography2002-12-22 20:35:132003-02-01 19:36:54Theorem% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $A_t$ 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 $x_{t+1}=A_tx_t$, 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 \PMlinkescapetext{dominant} eigenvalue of $A_\infty$ (also called the Perron-Frobenius eigenvector of $A_\infty$).