Open Access
2017 Note on A. Barbour’s paper on Stein’s method for diffusion approximations
Mikołaj J. Kasprzak, Andrew B. Duncan, Sebastian J. Vollmer
Electron. Commun. Probab. 22: 1-8 (2017). DOI: 10.1214/17-ECP54


In [2] foundations for diffusion approximation via Stein’s method are laid. This paper has been cited more than 130 times and is a cornerstone in the area of Stein’s method (see, for example, its use in [1] or [7]). A semigroup argument is used in [2] to solve a Stein equation for Gaussian diffusion approximation. We prove that, contrary to the claim in [2], the semigroup considered therein is not strongly continuous on the Banach space of continuous, real-valued functions on $D[0,1]$ growing slower than a cubic, equipped with an appropriate norm. We also provide a proof of the exact formulation of the solution to the Stein equation of interest, which does not require the aforementioned strong continuity. This shows that the main results of [2] hold true.


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Mikołaj J. Kasprzak. Andrew B. Duncan. Sebastian J. Vollmer. "Note on A. Barbour’s paper on Stein’s method for diffusion approximations." Electron. Commun. Probab. 22 1 - 8, 2017.


Received: 22 February 2017; Accepted: 4 April 2017; Published: 2017
First available in Project Euclid: 15 April 2017

zbMATH: 1381.60014
MathSciNet: MR3645505
Digital Object Identifier: 10.1214/17-ECP54

Primary: 60B10 , 60F17
Secondary: 60E05 , 60J60 , 60J65

Keywords: diffusion approximations , Donsker’s theorem , Stein’s method

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