A stochastic ordering approach is applied with Stein's method for approximation by the equilibrium distribution of a birth-death process. The usual stochastic order and the more general s-convex orders are discussed. Attention is focused on Poisson and translated Poisson approximations of a sum of dependent Bernoulli random variables, for example, k-runs in independent and identically distributed Bernoulli trials. Other applications include approximation by polynomial birth-death distributions.
"Stein's method and stochastic orderings." Adv. in Appl. Probab. 44 (2) 343 - 372, June 2012. https://doi.org/10.1239/aap/1339878715