Upper Bounds for Stein-Type Operators

Abstract

We present sharp bounds on the supremum norm of $\mathcal{D}^jSh$ for $j\geq2$, where $\mathcal{D}$ is the differential operator and $S$ the Stein operator for the standard normal distribution. The same method is used to give analogous bounds for the exponential, Poisson and geometric distributions, with $\mathcal{D}$ replaced by the forward difference operator in the discrete case. We also discuss applications of these bounds to the central limit theorem, simple random sampling, Poisson-Charlier approximation and geometric approximation using stochastic orderings.