Local Limit Approximations for Markov Population Processes

Abstract

In this paper we are concerned with the equilibrium distribution ∏ n of the nth element in a sequence of continuous-time density-dependent Markov processes on the integers. Under a (2+α)th moment condition on the jump distributions, we establish a bound of order O(n -(α+1)/2√logn) on the difference between the point probabilities of ∏ n and those of a translated Poisson distribution with the same variance. Except for the factor √logn, the result is as good as could be obtained in the simpler setting of sums of independent, integer-valued random variables. Our arguments are based on the Stein-Chen method and coupling.