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 the rate of convergence of the sum of the sample extremes
Download PDF
Download PDF
  • Published: October 1988

On the rate of convergence of the sum of the sample extremes

  • Joop Mijnheer1 

Probability Theory and Related Fields volume 79, pages 317–325 (1988)Cite this article

  • 91 Accesses

  • 1 Citations

  • Metrics details

Summary

Let X 1, X 2,..., X n be independent random variables having a common distribution in the domain of normal attraction of a completely asymmetric stable law with characteristic exponent α}(0,1) and support bounded below. Let X n:n ≧X n:n -1≧...≧X n:1 denote the ordered sample. We obtain the rate of convergence of n -1/α (X n:n +...+X n:n-k n+1 ) to the stable limit law as both n and k n »∞. As a consequence we obtain a representation of the sum X n:n +...+X n:n-k n+1 .

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs and mathematical tables. National Bureau of Standards, Applied Mathematics Series 35 (1964)

  2. Arov, D.Z., Bobrov, A.A.: The extreme terms of a sample and their role in the sum of independent variables. Th. Probab. Appl. V, 377–395 (1960)

    Google Scholar 

  3. Cramér, H.: On asymptotic expansions for sums of independent random variables with a limiting stable distribution. Sankhya A 25, 13–24 (1963)

    Google Scholar 

  4. Csörgö, S., Horváth, L., Mason, D.M.: What portion of the sample makes a partial sum asymptotically stable or normal. Probab. Th. Rel. Fields 72, 1–16 (1986)

    Google Scholar 

  5. Darling, D.A.: The influence of the maximum term in the addition of independent random variables. Trans. Am. Math. Soc. 73, 95–107 (1952)

    Google Scholar 

  6. Hall, P.: On the extreme terms of a sample from the domain of attraction of a stable law. J. London Math. Soc. 18, 181–191 (1978)

    Google Scholar 

  7. Laha, R.G., Rohatgi, V.K.: Probability theory. New York: Wiley 1979

    Google Scholar 

  8. Mijnheer, J.L.: Limit theorems for sums of random variables with a stable limit law. In: Prohorov, Yu.V. (ed.). Proceedings IV Vilnius conference, vol. II, pp. 253–262 Utrecht: V.N.U. Science Press 1986

    Google Scholar 

  9. Petrov, V.V.: Sums of independent random variables. Berlin Heidelberg New York: Springer 1975

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Leiden, 2300 RA, Leiden, The Netherlands

    Joop Mijnheer

Authors
  1. Joop Mijnheer
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mijnheer, J. On the rate of convergence of the sum of the sample extremes. Probab. Th. Rel. Fields 79, 317–325 (1988). https://doi.org/10.1007/BF00342230

Download citation

  • Received: 05 June 1987

  • Revised: 05 April 1988

  • Issue Date: October 1988

  • DOI: https://doi.org/10.1007/BF00342230

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

Keywords

  • Stochastic Process
  • Probability Theory
  • Statistical Theory
  • Independent Random Variable
  • Stable Limit
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