Journal of the Australian Mathematical Society - Series B
Vol. 40 Part 2 (1998)

Large deviations and quasi-stationarity for density-dependent birth-death processes

Terence Chan
Department of Actuarial Mathematics and Statistics
Heriot-Watt University
EH14 4AS U.K.


Consider a density-dependent birth-death process XN on a finite state 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 $\Pi_N$ 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 $\Pi_N$ is obtained by elementary explicit computations. However, when the birth-death process has an absorbing state at 0 (so $\Pi_N$ 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.

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