On a functional central limit theorem for Markov population processes

Let {XN(t)} be a sequence of continuous time Markov population processes on an n-dimensional integer lattice, such that XNhas initial state Nx(0) and has a finite number of possible transitions Jfrom any state X: let the transition X? X + Jhave rate NgJ(N–1X), and let gJ(x) and x(0) be fixed as Nvaries. The rate of convergence of vN(N–1XN(t) — ?(t)) to a Gaussian diffusion is investigated, where ?(t) is the deterministic approximation to N–1XN(t), and a method of deriving higher order asymptotic expansions for its distribution is justified. The methods are applied to two birth and death processes, and to the closed stochastic epidemic.