### Abstract

Consider a density-dependent birth-death process X_{N} on a finite state space of size N. Let P_{N} be the law (on D([0, T]) where T > 0 is arbitrary) of the density process X_{N}/N and let ?_{N} be the unique stationary distribution (on [0,1]) of X_{N}/N, if it exists. Typically, these distributions converge weakly to a degenerate distribution as N ? 8, so the probability of sets not containing the degenerate point will tend to 0; large deviations is concerned with obtaining the exponential decay rate of these probabilities. Friedlin-Wentzel theory is used to establish the large deviations behaviour (as N ? 8)of P_{N}. In the one-dimensional case, a large deviations principle for the stationary distribution ?_{N} is obtained by elementary explicit computations. However, when the birth-death process has an absorbing state at 0 (so ?_{N} no longer exists), the same elementary computations are still applicable to the quasi-stationary distribution, and we show that the quasi-stationary distributions obey the same large deviations principle as in the recurrent case. In addition, we address some questions related to the estimated time to absorption and obtain a large deviations principle for the invariant distribution in higher dimensions by studying a quasi-potential. © Australian Mathematical Society, 1998.

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

Pages (from-to) | 238-256 |

Number of pages | 19 |

Journal | Journal of the Australian Mathematical Society: Series B, Applied Mathematics |

Volume | 40 |

Issue number | 2 |

Publication status | Published - Oct 1998 |