Note on A. Barbour’s paper on Stein’s method for diffusion approximations

Journal article

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.

Authors: Kasprzak, M; Duncan, A; Vollmer, S

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Institute of Mathematical Statistics
• Journal: Electronic Communications in Probability
• Volume: 22
• Issue: 2017
• Pages: 1-8
• Publication date: 2017-04-15
• Acceptance date: 2017-04-04
• DOI: 10.1214/17-ECP54
• ISSN: 1083-589X
• Keywords: Donsker’s theorem; Stein’s method; diffusion approximations