# fundamental theorem of demography

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

$$\underset{t\to \mathrm{\infty}}{lim}\frac{{x}_{t}}{\parallel {x}_{t}\parallel}=p$$ |

where $p$ is the normalized ($\parallel p\parallel =1$) eigenvector^{} associated to the eigenvalue^{} of ${A}_{\mathrm{\infty}}$ (also called the Perron-Frobenius eigenvector of ${A}_{\mathrm{\infty}}$).

Title | fundamental theorem of demography |
---|---|

Canonical name | FundamentalTheoremOfDemography |

Date of creation | 2013-03-22 13:18:28 |

Last modified on | 2013-03-22 13:18:28 |

Owner | jarino (552) |

Last modified by | jarino (552) |

Numerical id | 6 |

Author | jarino (552) |

Entry type | Theorem |

Classification | msc 37A30 |

Classification | msc 92D25 |

Synonym | a weak ergodic theorem |

Related topic | PerronFrobeniusTheorem |