Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
On smoothness conditions and convergence rates in the CLT in Banach spaces
Download PDF
Download PDF
  • Published: June 1993

On smoothness conditions and convergence rates in the CLT in Banach spaces

  • Vidmantas Bentkus1 &
  • Friedrich Götze2 

Probability Theory and Related Fields volume 96, pages 137–151 (1993)Cite this article

  • 176 Accesses

  • 1 Citations

  • Metrics details

Summary

In Banach spaces the rate of convergence in the Central Limit Theorem is of orderO(n−1/2) for sets which have ‘regular’ boundaries with respect to the given covariance structure and which are three times differentiable. We show that in infinite dimensional spaces it is impossible to weaken this differentiability condition in general, whereas in finite dimensional spaces the assumption of convexity suffices. Similar results hold for the expectation of smooth functionals.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  • Alekseev, V.M., Tikhomirov, V.M., Fomin, S.V.: Optimal control. New York London: Consultants Bureau 1987

    Google Scholar 

  • Aliev, F.A.: A lower bound for the convergence rate in the CLT in Hilbert space. Theory Probab. Appl.31, 730–733 (1987)

    Google Scholar 

  • Barsov, S.S.: Rates of convergence to the normal distribution and decrease of the tail of the summand distribution. Theory Probab. Appl.32, 329–331 (1987)

    Google Scholar 

  • Bentkus, V.: Lower bounds for the sharpness of a normal approximation in Banach spaces. Lith. Math. J.24, 6–10 (1984)

    Google Scholar 

  • Bentkus, V.: Asymptotic expansions for moments in the CLT in Banach spaces. Lith. Math. J.26, 10–26 (1986a)

    Google Scholar 

  • Bentkus, V.: Dependence of the Berry-Esseen estimate on the dimension. Lith. Math. J.26, 110–113 (1986b)

    Google Scholar 

  • Bentkus, V.: Lower bounds for the rate of convergence in the central limit theorem in Banach spaces. Lith. Math. J.26, 312–319 (1986c)

    Google Scholar 

  • Bentkus, V.: Lower estimates of the convergence rate in the CLT in Banach spaces. Probab. Theory Math. Stat.1, 171–187 (1987)

    Google Scholar 

  • Bentkus, V., Götze, F., Paulauskas, V., Račkauskas, A.: The accuracy of Gaussian approximation in Banach spaces. SFB 343 Bielefeld, Preprint 90-100, Universität Bielefeld; in Russian: Itogi Nauki Teh., Sovr. Probl. Matem., Moskva, VINITI81, 39–139 (1991), in English in: Encycl. Math. Sci. Berlin Heidelberg New York: Springer (to appear)

  • Bentkus, V., Račkauskas, A.: Estimates of the rate of approximation of independent random variables in a Banach space. I. Lith. Math. J.23, 223–234 (1983). II. Lith. Math. J.23, 344–352 (1983)

    Google Scholar 

  • Bloznelis, M.: Lower bound for the rate of convergence in the CLT in Hilbert space. Lith. Math. J.29, 333–338 (1989)

    Google Scholar 

  • Borell, Ch.: Convex measures on locally convex spaces. Ark. Mat.12, 239–252 (1974)

    Google Scholar 

  • Borisov, I.S.: A remark on the speed of convergence in the central limit theorem in Banach spaces. Sib. Math. J.26, 180–185 (1985)

    Google Scholar 

  • Cartan, H.: Calcul differentiel. Formes differentielles. Paris: Hermann 1971

    Google Scholar 

  • Götze, F.: On Edgeworth expansions in Banach spaces. Ann. Probab.9, 852–859 (1981)

    Google Scholar 

  • Götze, F.: On the rate of convergence in the central limit theorem in Banach spaces. Ann. Probab.14, 922–942 (1986)

    Google Scholar 

  • Götze, F.: Edgeworth expansions in functional limit theorems. Ann. Probab.17, 1602–1634 (1989)

    Google Scholar 

  • Kuelbs, J., Kurtz, T.: Berry-Esseen estimates in Hilbert space and an application to the law of iterated logarithm. Ann. Probab.2, 387–407 (1974)

    Google Scholar 

  • Ledoux, M., Talagrand, M.: Probability in Banach spaces. Isoperimetry and processes. Berlin Heidelberg New York: Springer 1991

    Google Scholar 

  • Nagaev, S.V.: An estimate of the Berry-Esseen type for sums of Hilbert space valued r.v.'s. (in Russian). Sib. Math. J.30, 84–96 (1989)

    Google Scholar 

  • Osipov, L.V., Rotar, V.I.: On an infinite-dimensional central limit theorem. Theory Probab. Appl.29, 375–382 (1985)

    Google Scholar 

  • Paulauskas, V.: On the rate of convergence in the central limit theorem in certain Banach spaces. Theory Probab. Appl.21, 754–769 (1976)

    Google Scholar 

  • Paulauskas, V., Račkauskas, A.: Aproximation, theory in the central limit theorem. Exact results in Banach spaces. Dordrecht: Kluwer 1989; in Russian: Vilnius: Mokslas 1987

    Google Scholar 

  • Rachev, S.T., Yukich, J.E.: Rates for the CLT via new ideal metrics. Ann. Probab.17, 775–788 (1989)

    Google Scholar 

  • Račkauskas, A.: On the convergence rate in martingale CLT in Hilbert space. Preprint No 031. SFB 343 at Bielefeld, Univ. Bielefeld (1990)

  • Rhee, W.S., Talagrand, M.: Bad rates of convergence for the CLT in Hilbert space. Ann. Probab.12, 843–850 (1984)

    Google Scholar 

  • Sazonov, V.V., Ul'yanov, V.V., Zalesskii, B.A.: A precise estimate of the rate of convergence in the CLT in Hilbert space. Prepr., Stekov Math. Inst. USSR Acad. Sci.6 (1989)

  • Senatov, V.V.: Some lower convergence rate estimates in the CLT in Hilbert space. Sov. Math. Dokl.23, 188–192 (1981)

    Google Scholar 

  • Yurinskii, V.V.: On the accuracy of normal approximation of the probability of hitting a ball. Theory Probab. Appl.27, 280–289 (1982)

    Google Scholar 

  • Zalesskii, B.A.: On the accuracy of normal approximation in Banach spaces. Theory Probab. Appl.33, 239–247 (1988)

    Google Scholar 

  • Zolotarev, V.M.: Approximation of distributions of sums of independent random variables with values in infinite-dimensional spaces. Theory Probab. Appl.21, 721–737 (1976)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Informatics, Akademijos 4, 2600, Vilnius, Lithyania

    Vidmantas Bentkus

  2. Fakultät für Mathematik, Universität Bielefeld, Postfach 8640, W-4800, Bielefeld 1, Germany

    Friedrich Götze

Authors
  1. Vidmantas Bentkus
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Friedrich Götze
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research supported by SFB 343 at Bielefeld and by the Alexander von Humboldt Foundation and completed at the University of Bielefeld, FRG

Research supported by the SFB 343 at Bielefeld

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bentkus, V., Götze, F. On smoothness conditions and convergence rates in the CLT in Banach spaces. Probab. Th. Rel. Fields 96, 137–151 (1993). https://doi.org/10.1007/BF01192130

Download citation

  • Received: 16 March 1992

  • Revised: 03 December 1992

  • Issue Date: June 1993

  • DOI: https://doi.org/10.1007/BF01192130

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Mathematics Subject Classification (1991)

  • 60B12
  • 60F10
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature