Birth–death processes form a natural class where ideas and results on large deviations can be tested. We derive a large-deviation principle under an assumption that the rate of jump down (death) grows asymptotically linearly with the population size, while the rate of jump up (birth) grows sublinearly. We establish a large-deviation principle under various forms of scaling of the underlying process and the corresponding normalization of the logarithm of the large-deviation probabilities. The results show interesting features of dependence of the rate functional upon the parameters of the process and the forms of scaling and normalization.

We study the quasi-stationary behavior of the birth–death process with an entrance boundary at infinity. We give by the h-transform an alternative and simpler proof for the exponential convergence of conditioned distributions to a unique quasi-stationary distribution in the total variation norm. In addition, we also show that starting from any initial distribution the conditional probability converges to the unique quasi-stationary distribution exponentially fast in the $\psi$ -norm.

We consider lumpability for continuous-time Markov chains and provide a simple probabilistic proof of necessary and sufficient conditions for strong lumpability, valid in circumstances not covered by known theory. We also consider the following marginalisability problem. Let {X{t)} = {(X1(t), X2(t), · ··, Xm(t))} be a continuous-time Markov chain. Under what conditions are the marginal processes {X1(t)}, {X2(t)}, · ··, {Xm(t)} also continuous-time Markov chains? We show that this is related to lumpability and, if no two of the marginal processes can jump simultaneously, then they are continuous-time Markov chains if and only if they are mutually independent. Applications to ion channel modelling and birth–death processes are discussed briefly.

An efficient computational approach to the analysis of finite birth-and-death models in a Markovian environment is given. The emphasis is upon obtaining numerical methods for evaluating stationary distributions and moments of first-passage times.

In this note we are concerned with the rate of extinction of certain continuous-time birth-death processes on the positive integers with absorption at 0. The class we deal with includes birth-death processes with mean holding time h(i) at i such that h (i)∼ i–α as i →∞, 0 ≦ α< 1. In general, our result estimates to within a constant multiple the probability of non-extinction by time t. For h(i)∼ i–α, the result states that the probability of non-extinction is of order t−1/(2-α) We give an application to interacting particle systems.