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.
Highlights
Stein’s method is a powerful technique for assessing the distributional distance between a random variable of interest W and a limit random variable Z
Stein [52] originally developed the method for normal approximation, and the method was adapted to Poisson approximation by Chen [7]
The second step of Stein’s method, which will be the focus of this paper, concerns the problem of obtaining a solution f to the Stein equation (1.2) and establishing estimates for f and some of its lower order derivatives
Summary
Stein’s method is a powerful technique for assessing the distributional distance between a random variable of interest W and a limit random variable Z. Stein [52] originally developed the method for normal approximation, and the method was adapted to Poisson approximation by Chen [7]
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