### Abstract

We consider a Markov chain X^{ε} _{n} obtained by adding small noise to a discrete time dynamical system and study the chain's quasi-stationary distribution (qsd). The dynamics are given by iterating a function f: I → I for some interval I when f has finitely many fixed points, some stable and some unstable. We show that under some conditions the quasi-stationary distribution of the chain concentrates around the stable fixed points when ε → 0. As a corollary, we obtain the result for the case when f has a single attracting cycle and perhaps repelling cycles and fixed points. In this case, the quasi-stationary distribution concentrates on the attracting cycle. The result applies to the model of population dependent branching processes with periodic conditional mean function.

Original language | English |
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Pages (from-to) | 300-315 |

Number of pages | 16 |

Journal | Annals of Applied Probability |

Volume | 8 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jan 1998 |

Externally published | Yes |

### Keywords

- Branching systems
- Large deviations
- Logistic map
- Quasi-stationary distribution

## Cite this

*Annals of Applied Probability*,

*8*(1), 300-315. https://doi.org/10.1214/aoap/1027961045