## Electronic Journal of Probability

### An iterative technique for bounding derivatives of solutions of Stein equations

#### Abstract

We introduce a simple iterative technique for bounding derivatives of solutions of Stein equations $Lf=h-\mathbb{E} h(Z)$, where $L$ is a linear differential operator and $Z$ is the limit random variable. Given bounds on just the solutions or certain lower order derivatives of the solution, the technique allows one to deduce bounds for derivatives of any order, in terms of supremum norms of derivatives of the test function $h$. This approach can be readily applied to many Stein equations from the literature. We consider a number of applications; in particular, we derive new bounds for derivatives of any order of the solution of the general variance-gamma Stein equation. Finally, we present a connection between Stein equations and Poisson equations, from which we first recognised the importance of the iterative technique to Stein’s method.

#### Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 96, 39 pp.

Dates
Accepted: 19 October 2017
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.ejp/1510802250

Digital Object Identifier
doi:10.1214/17-EJP118

Mathematical Reviews number (MathSciNet)
MR3724564

Zentralblatt MATH identifier
06827073

#### Citation

Döbler, Christian; Gaunt, Robert E.; Vollmer, Sebastian J. An iterative technique for bounding derivatives of solutions of Stein equations. Electron. J. Probab. 22 (2017), paper no. 96, 39 pp. doi:10.1214/17-EJP118. https://projecteuclid.org/euclid.ejp/1510802250

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