Translated Poisson approximation for Markov chains
  The paper is concerned with approximating the distribution of a sum W of n integer valued random variables Y_i, whose distributions depend on the state of an underlying Markov chain X. The approximation is in terms of a translated Poisson distribution, with mean and variance chosen to be close to those of W, and the error is measured with respect to the total variation norm. Error bounds comparable to those found for normal approximation with respect to the weaker Kolmogorov distance are established, provided that the distribution of the sum of the Y_i's between the successive visits of X to a reference state is aperiodic. Without this assumption, approximation in total variation cannot be expected to be good. 
A. D. Barbour and Torgny Lindvall