Equilibrium distributions Markov population processes

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Abstract
Distributional limit theorems, together with rates of convergence, are obtained for the equilibrium distributions of a wide variety of one-dimensional Markov population processes. Three separate cases are considered. First, in the standard setting, the convergence as N→∞ of √N(xN -c) to a normal distribution is established, together with a rate of convergence of O(N −1/2), under weaker conditions than those previously imposed: here, c represents the unique equilibrium of the deterministic equations ẋ = F(x), and xN denotes the population process under its equilibrium distribution. This convergence holds if F′(c)F′(c) = 0, both the normalization and the limit distribution are different. Finally, sequences of processes xN suitable for approximating genetical models are considered. In these circumstances, xN itself converges in distribution as N→∞, and the convergence rate is essentially O(N -1), though modification is sometimes needed near natural boundaries.