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 convergence rates of suprema
Download PDF
Download PDF
  • Published: December 1991

On convergence rates of suprema

  • Peter Hall1 

Probability Theory and Related Fields volume 89, pages 447–455 (1991)Cite this article

  • 318 Accesses

  • 60 Citations

  • 3 Altmetric

  • Metrics details

Summary

It is shown that the convergence rate of suprema of stationary Gaussian and related processes, such as processes defined by the empirical distribution function, is logarithmically slow, even if the rates are to be uniform over as few as three points. It is proved that the bootstrap approximation provides a substantial improvement.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Berman, S.M.: Asymptotic independence of the numbers of high and low level crossings of stationary Gaussian processes. Ann. Math. Stat.42, 927–945 (1971)

    Google Scholar 

  2. Berman, S.M.: Maxima and high level excursions of stationary Gaussian processes. Trans. Am. Math. Soc.160, 65–85 (1971)

    Google Scholar 

  3. Berman, S.M.: Maximum and high level excursion of a Gaussian process with stationary increments. Ann. Math. Stat.43, 1247–1266 (1972)

    Google Scholar 

  4. Bickel, P.J., Rosenblatt, M.: On some global measures of the deviations of density function estimates. Ann. Stat.1, 1071–1095 (1973)

    Google Scholar 

  5. Cramér, H., Leadbetter, M.R.: Stationary and related stochastic processes. New York: Wiley 1967

    Google Scholar 

  6. Fisher, R.A., Tippett, L.H.C.: Limiting forms of the frequency distribution of the largest or smallest member of a sample. Proc. Cambridge Philos. Soc.24, 180–190 (1928)

    Google Scholar 

  7. Hall, P.: The rate of convergence of normal extremes. J. Appl. Probab.16, 433–439 (1979)

    Google Scholar 

  8. Komlós, J., Major, P., Tusnády, G.: An approximation of partial sums of independent rvs, and the sample df. I. Z. Wahrscheinlichkeits theor. Verw. Geb.32, 111–131 (1975)

    Google Scholar 

  9. Konakov, V.D., Piterbarg, V.I.: Convergence rate of maximum deviation distribution of Gaussian processes and empirical densities, II. Theor. Probab. Appl27, 759–779 (1982)

    Google Scholar 

  10. Konakov, V.D., Piterbarg, V.I.: Rate of convergence of distribution of maximal deviations of Gaussian processes and empirical density functions, I. Theor. Probab. Appl.28, 172–178 (1983)

    Google Scholar 

  11. Konakov, V.D., Piterbarg, V.I.: On the convergence rate of maximal deviation distribution for kernel regression estimates. J. Multivariate Anal.15, 279–294 (1984)

    Google Scholar 

  12. Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and related properties of random sequences and processes. Berlin Heidelberg New York: Springer 1983

    Google Scholar 

  13. Pickands, J.: Asymptotic properties of the maximum in a stationary Gaussian process. Trans. Am. Math. Soc.145, 75–86 (1969)

    Google Scholar 

  14. Silverman, B.W.: Weak and strong uniform consistency of the kernel estimate of a density function and its derivatives. Ann. Stat.6, 177–184 (1978) (Addendum 1980, Ann. Stat.8, 1175–1176.)

    Google Scholar 

  15. Silverman, B.W.: Density estimation for statistics and data analysis. London: Chapman and Hall 1986

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, Australian National University, GPO Box 4, 2601, Canberra, ACT, Australia

    Peter Hall

Authors
  1. Peter Hall
    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

Hall, P. On convergence rates of suprema. Probab. Th. Rel. Fields 89, 447–455 (1991). https://doi.org/10.1007/BF01199788

Download citation

  • Received: 21 June 1989

  • Revised: 15 March 1991

  • Issue Date: December 1991

  • DOI: https://doi.org/10.1007/BF01199788

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

  • Distribution Function
  • Stochastic Process
  • Probability Theory
  • Convergence Rate
  • Mathematical Biology
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