## Electronic Communications in Probability

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

#### Abstract

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

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

Dates
Accepted: 4 April 2017
First available in Project Euclid: 15 April 2017

https://projecteuclid.org/euclid.ecp/1492221618

Digital Object Identifier
doi:10.1214/17-ECP54

Mathematical Reviews number (MathSciNet)
MR3645505

Zentralblatt MATH identifier
1381.60014

#### Citation

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

#### References

• [1] A.D. Barbour, Stein’s method and Poisson process convergence, Journal of Applied Probability 25 (1988), 175–184.
• [2] A.D. Barbour, Stein’s Method for Diffusion Approximations, Probability Theory and Related Fields 84 (1990), 297–322.
• [3] Giuseppe Daprato and Alessandra Lunardi, On the Ornstein-Uhlenbeck operator in spaces of continuous functions, Journal of Functional Analysis 131 (1995), no. 1, 94–114.
• [4] P. Doersek and J. Teichmann, A Semigroup Point of View On Splitting Schemes For Stochastic (Partial) Differential Equations, arXiv:1011.2651, 2010.
• [5] S.N. Ethier and T.G. Kurtz, Markov processes: characterization and convergence, Wiley, New York, 1986.
• [6] J. Gorham, A.B. Duncan, S.J. Vollmer, and L. Mackey, Measuring sample quality with diffusions, arXiv:1611.06972, 2016.
• [7] S. Holmes and G. Reinert, Stein’s method for bootstrap, Lecture Notes-Monograph Series, vol. 46, Institute of Mathematical Statistics, 2004.
• [8] Luigi Manca, Kolmogorov operators in spaces of continuous functions and equations for measures, Ph.D. thesis, Scuola Normale Superiore di Pisa, 2008.