Probability Theory and Related Fields

, Volume 84, Issue 3, pp 297–322 | Cite as

Stein's method for diffusion approximations

  • A. D. Barbour


Stein's method of obtaining distributional approximations is developed in the context of functional approximation by the Wiener process and other Gaussian processes. An appropriate analogue of the one-dimensional Stein equation is derived, and the necessary properties of its solutions are established. The method is applied to the partial sums of stationary sequences and of dissociated arrays, to a process version of the Wald-Wolfowitz theorem and to the empirical distribution function.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arratia, R., Goldstein, L., Gordon, L.: Two moments suffice for Poisson approximation: the Chen-Stein method. Ann. Probab.17, 9–25 (1989)Google Scholar
  2. 2.
    Barbour, A.D.: Stein's method and Poisson process convergence. J. Appl. Probab.25(A),175–184 (1988)Google Scholar
  3. 3.
    Barbour, A.D., Holst, L., Janson, S.: Poisson approximation (in preparation)Google Scholar
  4. 4.
    Billingsley, P.: Convergence of probability measures. New York: Wiley (1968)Google Scholar
  5. 5.
    Bolthausen, E.: An estimate of the remainder in a combinatorial central limit theorem. Z. Wahrscheinlichlichkeitstheor. Verw. Geb.66, 379–386 (1984)Google Scholar
  6. 6.
    Chen, L.H.Y.: Poisson approximation for dependent trails. Ann. Probab.3, 535–545 (1975)Google Scholar
  7. 7.
    Ethier, S.N., Kurtz, T.G.: Markov processes: characterization and convergence. New York: Wiley 1986Google Scholar
  8. 8.
    Götze, F.: (1989) On the rate of convergence in the multivariate CLT. Ann. Probab. (in press)Google Scholar
  9. 9.
    Hall, P. and Heyde, C.C.: Martingale limit theory and its application. New York: Academic Press 1980Google Scholar
  10. 10.
    Lamperti, J.: Probability. New York: Benjamin 1966Google Scholar
  11. 11.
    McGinley, W.G., Sibson, R.: Dissociated random variables. Math. Proc. Cam. Phil. Soc.77, 185–188 (1975)Google Scholar
  12. 12.
    Pollard, D.: Convergence of stochastic processes. Berlin Heidelberg New York: Springer 1984Google Scholar
  13. 13.
    Stein, C.: A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proc. Sixth Berkeley Symp. Math. Statist. Probab.2, 583–602 (1970)Google Scholar
  14. 14.
    Stein, C.: Approximate computation of expectations. (IMS Lecture Notes, vol. 7.) Hayward, Calif. (1986)Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • A. D. Barbour
    • 1
  1. 1.Institut für Angewandte MathematikUniversität ZürichZürichSwitzerland

Personalised recommendations