An iterative technique for bounding derivatives of solutions of Stein equations

TL;DR

This paper presents an iterative method to derive bounds on derivatives of solutions to Stein equations, facilitating analysis in probability approximations and applied statistics.

Contribution

The paper introduces a new iterative technique for bounding derivatives of Stein equation solutions, applicable to various equations including the variance-gamma case.

Findings

Effective bounds for derivatives of all orders of Stein solutions

Applicable to multiple Stein equations from the literature

Simplifies derivative bounding process in probabilistic analysis

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 bounds for derivatives of any order of the solution of the general variance-gamma Stein equation.