3 research outputs found
Local limit approximations for Markov population processes
The paper is concerned with the equilibrium distribution of the
-th element in a sequence of continuous-time density dependent Markov
processes on the integers. Under a (2+\a)-th moment condition on the jump
distributions, we establish a bound of order O(n^{-(\a+1)/2}\sqrt{\log n}) on
the difference between the point probabilities of and those of a
translated Poisson distribution with the same variance. Except for the factor
, 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.Comment: 19 page
Translated Poisson approximation to equilibrium distributions of Markov population processes
The paper is concerned with the equilibrium distributions of continuous-time
density dependent Markov processes on the integers. These distributions are
known typically to be approximately normal, and the approximation error, as
measured in Kolmogorov distance, is of the smallest order that is compatible
with their having integer support. Here, an approximation in the much stronger
total variation norm is established, without any loss in the asymptotic order
of accuracy; the approximating distribution is a translated Poisson
distribution having the same variance and (almost) the same mean. Our arguments
are based on the Stein-Chen method and Dynkin's formula.Comment: 18 page