## Abstract

This paper studies the limit behaviour of {(Z_{n} − a_{n})/b_{n}} where {Z_{n}} is a real-valued temporally homogeneous Markov chain, and {a_{n}} and {b_{n}} are some constants; the results are then applied to a general population model. In such a model Z_{n} represents the nth generation population size and is defined as [formulla omitted] where {X_{i}
^{n-1}} are the offspring variables of the (n−1)th generation which are assumed to depend on n, i and Z_{n−1}’ whereas the classical conditional independence of {x_{i}
^{n}, i=1,…,Z_{n}} given Z_{n} is superseded by milder assumptions. Some necessary and sufficient conditions for {Z_{n}/b_{n}} to converge a.s. are derived, and some results on the robustness of the asymptotic behaviour of the Galton-Watson process are obtained when offspring independence is relaxed.

Original language | English |
---|---|

Pages (from-to) | 283-307 |

Number of pages | 25 |

Journal | Stochastic Analysis and Applications |

Volume | 4 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Jan 1986 |

Externally published | Yes |