J Austral Math Soc Ser B 40 pp238--256, 1998.
(Received 17 May 1997; revised 28 October 1997)
Consider a density-dependent birth-death process XN on a finite space of size N. Let PN be the law (on D([0, T]) where T > 0 is arbitrary) of the density process XN / N and let \PiN be the unique stationary distribution (on [0, 1]) of XN / N, if it exists. Typically, these distributions converge weakly to a degenerate distribution as N \rightarrow \infty, 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 \rightarrow \infty) of PN . In the one-dimensional case, a large deviations principle for the stationary distribution \PiN is obtained by elementary explicit computations. However, when the birth-death process has an absorbing state at 0 (so \PiN 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.
Last Modified: Thu Jan 3 13:52:21 2002