Advertisement

Journal of Theoretical Probability

, Volume 31, Issue 3, pp 1590–1605 | Cite as

Approximate Central Limit Theorems

  • Ben Berckmoes
  • Geert Molenberghs
Article

Abstract

We refine the classical Lindeberg–Feller central limit theorem by obtaining asymptotic bounds on the Kolmogorov distance, the Wasserstein distance, and the parameterized Prokhorov distances in terms of a Lindeberg index. We thus obtain more general approximate central limit theorems, which roughly state that the row-wise sums of a triangular array are approximately asymptotically normal if the array approximately satisfies Lindeberg’s condition. This allows us to continue to provide information in nonstandard settings in which the classical central limit theorem fails to hold. Stein’s method plays a key role in the development of this theory.

Keywords

Kolmogorov metric Lindeberg–Feller central limit theorem Lindeberg index Prokhorov metric Stein’s method Wasserstein metric 

Mathematics Subject Classification (2010)

60F05 

References

  1. 1.
    Barbour, A.D., Chen, L.H.Y.: An introduction to Stein’s method. Singapore University Press, World Scientific Publishing Co. Pte. Ltd, Singapore, Hackensack, NJ (2005)CrossRefzbMATHGoogle Scholar
  2. 2.
    Barbour, A.D., Hall, P.: Stein’s method and the Berry–Esseen theorem. Aust. J. Stat. 26(1), 8–15 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berckmoes, B., Lowen, R., Van Casteren, J.: Distances on probability measures and random variables. J. Math. Anal. Appl. 374(2), 412–428 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Berckmoes, B., Lowen, R., Van Casteren, J.: An isometric study of the Lindeberg–Feller central limit theorem via Stein’s method. J. Math. Anal. Appl. 405(2), 484–498 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Berckmoes, B., Lowen, R., Van Casteren, J.: Stein’s method and a quantitative Lindeberg CLT for the Fourier transforms of random vectors. J. Math. Anal. Appl. 433(2), 1441–1458 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Billingsley, P.: Convergence of Probability Measures. Wiley Series in Probability and Statistics: Probability and Statistics, 2nd edn. Wiley, New York (1999)CrossRefzbMATHGoogle Scholar
  7. 7.
    Bolley, F.: Separability and completeness for the Wasserstein distance. In: Séminaire de probabilités XLI, Lecture Notes in Mathematics, vol. 1934, pp. 371–377. Springer, Berlin (2008)Google Scholar
  8. 8.
    Chen, L.H.Y., Goldstein, L., Shao, Q.-M.: Normal Approximation by Stein’s method. Probability and its Applications (New York). Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    Chen, L.H.Y., Shao, Q.-M.: A non-uniform Berry–Esseen bound via Stein’s method. Probab. Theory Relat. Fields 120(2), 236–254 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Osipov, L.V.: A refinement of Lindeberg’s theorem. Teor. Verojatnost. i Primenen. 11, 339–342 (1966). (Russian)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Feller, W.: On the Berry–Esseen theorem. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 10, 261–268 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Loh, W.Y.: On the normal approximation for sums of mixing random variables. Master Thesis, Department of Mathematics, University of Singapore (1975)Google Scholar
  13. 13.
    Lowen, R.: Index Analysis. Approach Theory at Work. Springer Monographs in Mathematics. Springer, London (2015)Google Scholar
  14. 14.
    Meckes, E.: On Stein’s method for multivariate normal approximation. In: High dimensional probability V: the Luminy volume, pp. 153–178, Institute of Mathematical Statistics Collection, 5, Institute of Mathematical Statistics, Beachwood, OH (2009)Google Scholar
  15. 15.
    Nourdin, I., Peccati, G., Réveillac, A.: Multivariate normal approximation using Stein’s method and Malliavin calculus. Ann. Inst. Henri Poincaré Probab. Stat. 46(1), 45–58 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Rachev, S.T.: Probability metrics and the stability of stochastic models. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. Wiley, Chichester (1991)zbMATHGoogle Scholar
  17. 17.
    Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence, RI (2003)zbMATHGoogle Scholar
  18. 18.
    Zolotarev, V.M.: Probability metrics. Teor. Veroyatnost. i Primenen. 28(2), 264–287 (1983). (Russian)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Universiteit AntwerpenAntwerpBelgium
  2. 2.Universiteit HasseltHasseltBelgium
  3. 3.KU LeuvenLeuvenBelgium

Personalised recommendations