Advertisement

A New Way to Obtain Estimates in the Invariance Principle

  • Alexander I. Sakhanenko
Conference paper
Part of the Progress in Probability book series (PRPR, volume 47)

Abstract

This paper presents a new simple method of obtaining estimates on rates of convergence in the invariance principle, which may be used in arbitrary separable linear spaces. This method is applied to one-dimensional and infinite dimensional martingales, among other examples.

Keywords

Random Vector Joint Distribution Independent Random Variable Random Element Invariance Principle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bentkus V. (1985) Asymptotic analysis of sums of independent random elements in Banach spaces. Doctor Dissertation, Vilnius.Google Scholar
  2. Bentkus V., Liubinskas K. (1987) Rates of convergence in the invariance principle in Banach spaces. Lith. Math. J., 27, 3, 205–213.MathSciNetMATHCrossRefGoogle Scholar
  3. Bentkus V., Rakauskas A. (1984) Estimates of the distances between sums of independent random elements in Banach spaces. Theory Probab. Appl., 29, 1, 50–65.CrossRefGoogle Scholar
  4. Berkes I., Philipp W. (1979) Approximation theorems for independent and weakly dependent random vectors. Ann. Probab., 7, 1, 29–54.MathSciNetMATHCrossRefGoogle Scholar
  5. Dobrushin R.L. (1970) Prescribing a system of random variables by conditional distributions. Theory Probab. Appl., 15, 3, 458–486.MathSciNetMATHCrossRefGoogle Scholar
  6. Einmahl U. (1987) A useful estimate in the multidimensional invariance principle. Probab. Th. Rel. Fields, 76, 81–101.MathSciNetMATHCrossRefGoogle Scholar
  7. Lindvall T. (1992) Lectures on the Coupling Method, Wiley, New York.MATHGoogle Scholar
  8. Loéve M. (1978) Probability theory, Springer-Verlag, New York.MATHGoogle Scholar
  9. Morrow G., Philipp W. (1982) An almost sure invariance principle for Hilbert space valued martingales. Trans. Amer. Math. Soc., 273, 231–251.MathSciNetMATHCrossRefGoogle Scholar
  10. Monrad D., Philipp W. (1991) Nearby variables with nearby conditional laws and a strong approximation theorem for Hilbert space valued martingales. Probab. Th. Rel. Fields, 88, 381–404.MathSciNetMATHCrossRefGoogle Scholar
  11. Paulauskas V. and Rackauskas A. (1989) Approximation theory in the central limit theorem. Exact results in Banach spaces, Kluwer Academic Publisher, Dordrecht.CrossRefGoogle Scholar
  12. Sakhanenko A. I. (1974) Estimates of the rate of convergence in the invariance principle. Soviet Math. Dokl., 15, 6, 1752–1755.MATHGoogle Scholar
  13. Sakhanenko A. I. (1987) Simple method of obtaining estimates in the in- variance principle. Lecture Notes in Math., Springer-Verlag, 1299, 430–443.MathSciNetGoogle Scholar
  14. Skorohod A.V. (1976) On a representation of random variables. Theory Probab. Appl., 21, 3, 628–632.CrossRefGoogle Scholar
  15. Zaitsev A.Yu. (1998) Multidimensional version of the results of Komlôs, Major, Tusnâdy for vectors with finite exponential moments. ESAIM: Probability and Statistics, 2, 41–108.MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Alexander I. Sakhanenko
    • 1
  1. 1.Departamento de MatemáticasUniversidad Autónoma Metropolitana - Iztapalapa Av. Michoacán y la Purísima s/nMéxico, D.F.Mexico

Personalised recommendations