PlanetMath: fundamental theorem of demography

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fundamental theorem of demography

(Theorem)

Let 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

$\displaystyle \lim_{t\to\infty} \frac{x_t}{\Vert x_t\Vert} =p$

where

is the normalized ( $\Vert p\Vert=1$ ) eigenvector associated to the dominant eigenvalue of $A_\infty$ (also called the Perron-Frobenius eigenvector of $A_\infty$ ).

"fundamental theorem of demography" is owned by jarino.

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See Also: Perron-Frobenius theorem

Other names:

a weak ergodic theorem

Attachments:: proof of fundamental theorem of demography (Proof) by aplant
example of fundamental theorem of demography (Example) by mathcam

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Cross-references: eigenvalue, eigenvector, primitive matrices, sequence
There are 2 references to this entry.

This is version 3 of fundamental theorem of demography, born on 2002-12-22, modified 2003-02-01.
Object id is 3813, canonical name is FundamentalTheoremOfDemography.
Accessed 3438 times total.

Classification:

AMS MSC:	37A30 (Dynamical systems and ergodic theory :: Ergodic theory :: Ergodic theorems, spectral theory, Markov operators)
	92D25 (Biology and other natural sciences :: Genetics and population dynamics :: Population dynamics )

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