Signed Poisson approximations for Markov chains

Consider a sum of Markov dependent lattice variables. The normal approximation is trivial for this sum if the total variation distance is considered. Replacement of the normal approximation by its Poisson structured analogue changes the situation radically. Moreover, considering the Markov binomial distribution we prove that signed Poisson approximation can be more accurate than both the normal and Poisson approximations. Possible improvements due to asymptotic expansions are discussed.