Electronic Communications in Probability

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

Mikołaj J. Kasprzak, Andrew B. Duncan, and Sebastian J. Vollmer

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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.

Article information

Electron. Commun. Probab., Volume 22 (2017), paper no. 23, 8 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B10: Convergence of probability measures 60F17: Functional limit theorems; invariance principles
Secondary: 60J60: Diffusion processes [See also 58J65] 60J65: Brownian motion [See also 58J65] 60E05: Distributions: general theory

Stein’s method Donsker’s theorem diffusion approximations

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Kasprzak, Mikołaj J.; Duncan, Andrew B.; Vollmer, Sebastian J. Note on A. Barbour’s paper on Stein’s method for diffusion approximations. Electron. Commun. Probab. 22 (2017), paper no. 23, 8 pp. doi:10.1214/17-ECP54. https://projecteuclid.org/euclid.ecp/1492221618

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