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# 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 dominant eigenvalue of $A_{\infty}$ (also called the Perron-Frobenius eigenvector of $A_{\infty}$).

Related:

PerronFrobeniusTheorem

Synonym:

a weak ergodic theorem

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

37A30*no label found*92D25

*no label found*

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