Skip to main content
SpringerLink
Log in

Large Deviations of Empirical Processes

  • Conference paper
High Dimensional Probability III

Part of the book series: Progress in Probability ((PRPR,volume 55))

Abstract

We give necessary and sufficient conditions for the large deviations of empirical processes and of Banach space valued random vectors. We also consider the large deviations of partial sums processes. The main tool used is an isoperimetric inequality for empirical processes due to Talagrand.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M.A. ArconesThe large deviation principle for stochastic processes I. To appear in Theor. Probab. Applic. (2002).

    Google Scholar 

  2. M.A. ArconesThe large deviation principle for stochastic processes II. To appear in Theor. Probab. Applic. (2002).

    Google Scholar 

  3. R. R. BahadurSome Limit Theorems in Statistics. SIAM, Philadelphia, PA. 1971.

    Book  MATH  Google Scholar 

  4. R. R. Bahadur, and S. L. ZabellLarge deviations of the sample mean in general vector spaces. Ann. Probab. 57 (1979), 587–621.

    Article  MathSciNet  Google Scholar 

  5. A. A. BorovkovBoundary—value problems for random walks and large deviations in function spaces. Theor. Probab. Appl. 12 (1967), 575–595.

    MATH  Google Scholar 

  6. J. M. Borwein and A. S. LewisConvex Analysis and Nonlinear Optimization: Theory and Examples. Springer—Verlag, New York. 2000.

    MATH  Google Scholar 

  7. H. ChernoffA measure of asymptotic efficiency for tests of hypothesis based on the sum of observations. Ann. Math. Statist. 23 (1952), 493–507.

    Article  MathSciNet  MATH  Google Scholar 

  8. H. CramérSur un nouveau théorèm limite de la théorie des probabilities. Actualités Sci. Ind. 736 (1938), 5–23.

    Google Scholar 

  9. A. Dembo, and O. Zeitouni, (1999).Large Deviations Techniques and Applications. 2nd Edition, Springer, New York.

    Google Scholar 

  10. J. D. Deuschel, and D. W. StroockLarge Deviations. Academic Press, Inc., Boston, MA. 1989.

    MATH  Google Scholar 

  11. J. DiestelSequences and Series in Banach Spaces. Springer—Verlag, New York. 1984.

    Book  Google Scholar 

  12. M. D. Donsker, and S. R. S. VaradhanAsymptotic evaluation of certain Markov process expectations for large time III. Comm. Pure Appl. Math. 29 (1976), 389–461.

    Article  MathSciNet  MATH  Google Scholar 

  13. R. M. Dudley, Uniform Central Limit Theorems. Cambridge University Press, Cambridge. 1999.

    Book  MATH  Google Scholar 

  14. R. S. EllisLarge deviations for a general class of random vectors. Ann. Probab. 12 (1984), 1–12.

    Article  MathSciNet  MATH  Google Scholar 

  15. E. Giné, and J. ZinnLectures on the central limit theorem for empirical processes. Lect. Notes in Math. 1221 (1986), 50–112. Springer—Verlag, New York.

    Article  Google Scholar 

  16. P. Groeneboom; J. Oosterhoff and F. H. RuymgaartLarge deviation theorems for empirical probability measures. Ann. Probab. 7 (1979), 553–586.

    Article  MathSciNet  MATH  Google Scholar 

  17. T. Jiang; M. Bhaskara Rao, and X. P. WangLarge deviations for moving average processes. Stoch. Proc. Appl. 59 (1995), 309–320.

    Article  MATH  Google Scholar 

  18. M. Ledoux, and M. TalagrandProbability in Banach Spaces. Springer—Verlag, New York. 1991.

    MATH  Google Scholar 

  19. A. A. MogulskiiLarge deviations for trajectories of multidimensional random walks. Theor. Probab. Appl. 21 (1976), 300–315.

    Article  Google Scholar 

  20. J. SethuramanOn the probability of large deviations of families of sample means. Ann. Math. Statist. 35 (1964), 1964 1304–1316.

    Article  MathSciNet  MATH  Google Scholar 

  21. J. SethuramanOn the probability of large deviations of of mean for random variables in D[0,1]. Ann. Math. Statist. 36 (1965), 280–285.

    Article  MathSciNet  MATH  Google Scholar 

  22. J. SethuramanCorrections to: “On the probability of large deviations of families of sample means”. Ann. Math. Statist. 41 (1970), 1376–1380.

    Article  MathSciNet  MATH  Google Scholar 

  23. M. TalagrandNew concentration inequalities in product spaces. Inv. Math. 126 (1996), 505–563.

    Article  MathSciNet  MATH  Google Scholar 

  24. S.R.S. VaradhanLarge Deviations and Applications. SIAM, Philadelphia, Pennsylvania. (1984).

    Book  Google Scholar 

  25. L. WuLarge deviations, moderate deviations and LIL for empirical processes. Ann. Probab. 22 (1994), 17–27.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Binghamton University, Binghamton, NY, 13902, USA

    Miguel A. Arcones

Authors
  1. Miguel A. Arcones
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, University of Aarhus, Building 530 Ny Munkegade, Arhus C, 8000, Denmark

    Jørgen Hoffmann-Jørgensen

  2. Department of Statistics, University of Washington, Box 354322, Seattle, WA, 98195-4322, USA

    Jon A. Wellner

  3. Department of Mathematics, City College, New York, NY, 10031, USA

    Michael B. Marcus

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer Basel AG

About this paper

Cite this paper

Arcones, M.A. (2003). Large Deviations of Empirical Processes. In: Hoffmann-Jørgensen, J., Wellner, J.A., Marcus, M.B. (eds) High Dimensional Probability III. Progress in Probability, vol 55. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8059-6_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8059-6_13

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9423-4

  • Online ISBN: 978-3-0348-8059-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation