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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1492221618

Digital Object Identifier
doi:10.1214/17-ECP54

Subjects
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

Keywords
Stein’s method Donsker’s theorem diffusion approximations

Rights
Creative Commons Attribution 4.0 International License.

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


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References

  • [1] A.D. Barbour,Stein’s method and Poisson process convergence, Journal of Applied Probability25(1988), 175–184.
  • [2] A.D. Barbour,Stein’s Method for Diffusion Approximations, Probability Theory and Related Fields84(1990), 297–322.
  • [3] Giuseppe Daprato and Alessandra Lunardi,On the Ornstein-Uhlenbeck operator in spaces of continuous functions, Journal of Functional Analysis131(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.