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
A new ideal metric with applications to multivariate stable limit theorems
Download PDF
Download PDF
  • Published: June 1992

A new ideal metric with applications to multivariate stable limit theorems

  • S. T. Rachev1 &
  • L. Rüschendorf2 

Probability Theory and Related Fields volume 94, pages 163–187 (1992)Cite this article

  • 134 Accesses

  • 7 Citations

  • Metrics details

Summary

A new ideal metric of orderr>1 is introduced on ℝk and a thorough analysis of its metric properties is given. In comparison to the known ideal metric of Zolotarev this new metric allows estimates from above by pseudo difference moments and thus allows applications to stable limit theorems. As applications we give the right order Berry-Esséen type result in the stable case, obtain the limiting behaviour of multivariate summability methods and discuss the approximation problem by compound Poisson distributions.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Cuesta-Albertos, J.A., Rüschendorf, L. and Tuero-Diaz, A.: Optimal coupling of multivariate distributions and stochastic processes. (Preprint 1991)

  2. Gerber, H.: An introduction to mathematical risk theory. (Huebner Foundation Monograph, vol. 8) Philadelphia: University of Pennsylvania 1979

    Google Scholar 

  3. Hipp, C., Michel, R.: Risikotheorie: Stochastische Modelle und Statistische Methoden. DGVM, Heft 24 (1990)

  4. Ledoux, M., Talagrand, M.: Probability in banach spaces. Berlin Heidelberg New York: Springer 1991

    Google Scholar 

  5. Maejima, M.: Some limit theorems for summability methods of lid random variables. In: Kalashniko, V.V., et al., (eds.) Stability problems for stochastic models. Varna 1985. (Lect. Notes Math., vol. 1233, pp. 57–68) Berlin Heidelberg New York: Springer 1987

    Google Scholar 

  6. Maejima, M., Rabhev, S.T.: An ideal metric and the rate of convergence to a self similar process. Ann. Probab.15, 708–727 (1987)

    Google Scholar 

  7. Rachev, S.T., Rüschendorf, L.: Approximation of sums by compound Poisson distributions w.r.t. stop-loss distances. Adv. Appl. Probab22, 350–374 (1990)

    Google Scholar 

  8. Rachev, S.T., Rüschendorf, L.: A transformation property of minimal metrics. Theory Probab. Appl.35, 131–136 (1990)

    Google Scholar 

  9. Rachev, S.T.: Probability metrics and the stability of stochastic models. New York: Wiley 1991

    Google Scholar 

  10. Rüschendorf, L., Rachev, S.T.: A characterization of minimumL 2-distance. J. Multivariate Anal.32, 48–54 (1990)

    Google Scholar 

  11. Samorodnitskii, G. and Taqqu, M.S.: Stable random processes. Monograph (in preparation, 1991)

  12. Senatov, V.V.: Uniform estimates of the rate of convergence in the multidimensional central limit theorem. Theory Probab. Appl.25, 745–759 (1980)

    Google Scholar 

  13. 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 

  14. Zolotarev, V.M.: Ideal metrics in the problem of approximating distributions of sums of independent random variables. Theory Probab. Appl.22, 433–439 (1977)

    Google Scholar 

  15. Zolotarev, V.M.: Probability metrics. Theory Probab. Appl.28, 278–302 (1983)

    Google Scholar 

  16. Zolotarev, V.M.: Modern theory of summation of independent random variables (in Russian). Moscow: Nauka 1987

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Statistics and Applied Probability Program, University of California, 93106, Santa Barbara, CA, USA

    S. T. Rachev

  2. Institut für Mathematische Statistik, Westfälische Wilhelms-Universität, Einsteinstrasse 62, W-4400, Münster, Federal Republic of Germany

    L. Rüschendorf

Authors
  1. S. T. Rachev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. Rüschendorf
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research supported by NATO GRANT CRG 900 798 and by a DFG Grant

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Rachev, S.T., Rüschendorf, L. A new ideal metric with applications to multivariate stable limit theorems. Probab. Th. Rel. Fields 94, 163–187 (1992). https://doi.org/10.1007/BF01192443

Download citation

  • Received: 31 May 1991

  • Revised: 24 December 1991

  • Issue Date: June 1992

  • DOI: https://doi.org/10.1007/BF01192443

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)

  • 60 F 05
  • 60 E 15
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