On Stein factors and the construction of examples with sharp rates in Stein's method

The application of Stein's method for distributional approximation often involves so called Stein factors (also called magic factors) in the bound of the solutions to Stein equations. However, these factors sometimes contain additional terms such as a logarithmic term for Poisson point process approximation, leading to unsatisfactory estimates. Despite the fact that is has been shown for many of these Stein factors that the known bounds are sharp and thus that the additional terms cannot be avoided in general, no probabilistic examples have been presented in the literature, which justify these Stein factors. In this article we close this gap by constructing such examples more or less explicitly. As a side effect, a new interpretation of the solutions to Stein equations is given.