It is shown that the method of exchangeable pairs introduced by Stein [Approximate Computation of Expectations (1986) IMS, Hayward, CA] for normal approximation can effectively be used for translated Poisson approximation. Introducing an additional smoothness condition, one can obtain approximation results in total variation and also in a local limit metric. The result is applied, in particular, to the anti-voter model on finite graphs as analyzed by Rinott and Rotar [Ann. Appl. Probab. 7 (1997) 1080–1105], obtaining the same rate of convergence, but now for a stronger metric.
"Translated Poisson approximation using exchangeable pair couplings." Ann. Appl. Probab. 17 (5-6) 1596 - 1614, October 2007. https://doi.org/10.1214/105051607000000258