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fundamental theorem of demography
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(Theorem)
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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 $x_{t+1}=A_tx_t$ 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 dominant eigenvalue of $A_\infty$ (also called the
Perron-Frobenius eigenvector of $A_\infty$ .
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"fundamental theorem of demography" is owned by jarino.
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Cross-references: 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 4040 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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Pending Errata and Addenda
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