Pseudo-Poisson approximation for Markov chains

We consider the problem of approximating the distribution of a Markov chain with ‘rare’ transitions in an arbitrary state space by that of the corresponding pseudo-Poisson process. Sharp estimates for both first- and second-order approximations are obtained. The remarkable fact is that the convergence rate in this setup can be better than that in the ordinary Poisson theorem: the ergodicity of the embedded ‘routing’ Markov chain improves essentially the degree of approximation. This is of particular importance if the accumulated transition intensity of the chain is of a moderate size so that neither the usual estimates from the Poisson theorem nor the existence of a stationary distribution alone provide good approximation results. On the other hand, the estimates also improve the known results in the ordinary Poisson theorem.