Compound Poisson approximation is a useful tool in a variety of applications, including insurance mathematics, reliability theory, and molecu- lar sequence analysis. In this paper, we review the ways in which Stein’s method can currently be used to derive bounds on the error in such approximations. The theoretical basis for the construction of error bounds is systematically discussed, and a number of specific examples are used for illustration.We give no numerical comparisons in this paper, contenting ourselves with references to the literature, where compound Poisson approximations derived using Stein’s method are shown frequently to improve upon bounds obtained from problem specific, ad hoc methods.
"Compound Poisson approximation: a user's guide." Ann. Appl. Probab. 11 (3) 964 - 1002, August 2001. https://doi.org/10.1214/aoap/1015345355