LOCAL LIMIT APPROXIMATIONS FOR MARKOV POPULATION PROCESSES

In this paper we are concerned with the equilibrium distribution Pi(n) of the nth element in a sequence of continuous-time density-dependent Markov processes on the integers. Under a (2 + alpha)th moment condition on the jump distributions, we establish a bound of order O(n(-(alpha+1)/2) root log n) on the difference between the point probabilities of Pi(n) and those of a translated Poisson distribution with the same variance. Except for the factor root log n, 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.