Electronic Journal of Probability

Upper Bounds for Stein-Type Operators

Fraser Daly

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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.

Article information

Source
Electron. J. Probab., Volume 13 (2008), paper no. 20, 566-587.

Dates
Accepted: 12 April 2008
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819093

Digital Object Identifier
doi:10.1214/EJP.v13-479

Mathematical Reviews number (MathSciNet)
MR2399291

Zentralblatt MATH identifier
1196.60032

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 62E17: Approximations to distributions (nonasymptotic)

Keywords
Stein-type operator Stein's method central limit theorem Poisson-Charlier approximation stochastic ordering

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Daly, Fraser. Upper Bounds for Stein-Type Operators. Electron. J. Probab. 13 (2008), paper no. 20, 566--587. doi:10.1214/EJP.v13-479. https://projecteuclid.org/euclid.ejp/1464819093


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