Electronic Journal of Probability

An iterative technique for bounding derivatives of solutions of Stein equations

Christian Döbler, Robert E. Gaunt, and Sebastian J. Vollmer

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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

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

Received: 10 October 2016
Accepted: 19 October 2017
First available in Project Euclid: 16 November 2017

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 35A24: Methods of ordinary differential equations

Stein’s method Stein equation differential equation Poisson equation variance-gamma distribution

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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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