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
Laplace approximations for sums of independent random vectors
Download PDF
Download PDF
  • Published: October 1987

Laplace approximations for sums of independent random vectors

Part II. Degenerate maxima and manifolds of maxima

  • E. Bolthausen1 

Probability Theory and Related Fields volume 76, pages 167–206 (1987)Cite this article

  • 192 Accesses

  • 31 Citations

  • Metrics details

Summary

We consider expressions of the form Z n =E(exp nF(S n /n)) where S n is the sum of n i.i.d. random vectors with values in a Banach space and F is a smooth real valued function. By results of Donsker-Varadhan and Bahadur-Zabell one knows that lim (1/n) log Z n =sup x F(x)-h(x) where h is the so-called entropy function. In an earlier paper a more precise evaluation of Z n is given in the case where there was a unique point maximizing F-h and the curvature at the maximum was nonvanishing. The present paper treats the more delicate problem where these conditions fail to hold.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Azencott, R.: Grandes déviations et applications. Ecole d'été Saint-Flour VIII 1978. Lect. Notes Math. 774 (1980)

  2. Azencott, R.: Formule de Taylor stochastique et développement asymptotique d'intégrales de Feynman. Sém de Probabilités XVI, Supplément; Géométrie différentielle stochastique. Lect. Notes Math. 921, 237–285 (1982)

    Google Scholar 

  3. Bahadur, R.R., Zabell, S.L.: Large deviations of the sample mean in general vector spaces. Ann. Probab. 7, 537–621 (1979)

    Google Scholar 

  4. Bhattacharya, R.N., Rao, R.N.: Normal approximations and asymptotic expansions. New York: Wiley 1976

    Google Scholar 

  5. Bolthausen, E.: Laplace approximations for sums of independent random vectors. Probab. Th. Rel. Fields 72, 305–318 (1986)

    Google Scholar 

  6. Bolthausen, E.: Laplace approximations for Markov process expectations. To appear in Ann. Probab. (1986)

  7. Borell, C.: Gaussian radon measures on locally convex spaces. Math. Scand. 38, 265–284 (1976)

    Google Scholar 

  8. Csiszar, I.: I-divergence geometry of probability distributions and minimization problems. Ann. Probab. 3, 146–158 (1975)

    Google Scholar 

  9. Dawson, D.: Critical dynamics and fluctuations for a mean-field model of cooperative behavior. J. Stat. Phys. 31, 29–85 (1983)

    Google Scholar 

  10. Dawson, D., Gärtner, J.: Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Preprint (1987)

  11. de Acosta, A., Giné F.: Convergence of moments and related functionals in the general central limit theorem in Banach spaces. Z. Wahrscheinlichkeitstheor. Verw. Geb. 48, 213–231 (1979)

    Google Scholar 

  12. de Acosta, A.: Upper bounds for large deviations of dependent random vectors. Z. Wahrscheinlich-keitstheor. Verw. Geb. 69, 551–565 (1985)

    Google Scholar 

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

    Google Scholar 

  14. Ellis, R.S., Newman, C.M.: Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrscheinlichkeitstheor. Verw. Geb. 44, 117–139 (1978)

    Google Scholar 

  15. Ellis, R.S., Rosen, J.S.: Asymptotic analysis of Gauss integrals I, II: Trans. Am. Math. Soc. 273, 447–481 (1982) and Comm. Math. Phys. 82, 153–181 (1982)

    Google Scholar 

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

    Google Scholar 

  17. Jain, N.C.: Central limit theorem in Banach spaces. Proc. First Conf. on Prob. in Banach Spaces 1975. Lect. Notes in Math. 598 (1976)

  18. Kuo, H.-H.: Gaussian measures in Banach spaces. Lect. Notes Math. 463 (1975)

  19. Kusuoka and Tamura: The convergence of Gibbs measures associated with mean field potentials. J. Fac. Sci. Univ. Tokyo, Sect. 1A, 31, 223–245 (1984)

    Google Scholar 

  20. Martin-Löf, A.: Laplace approximation for sums of independent random variables. Z. Wahrschein-lichkeitstheor. Verw. Geb. 59, 101–115 (1982)

    Google Scholar 

  21. Messer, J., Spohn, H.: Statistical mechanics of the isothermal Lane-Emden equation. J. Stat. Phys. 29, 561–578 (1982)

    Google Scholar 

  22. Ney, P.: Dominating points and the asymptotics of large deviations for random walk on ℝd. Ann. Probab. 11, 158–167 (1983)

    Google Scholar 

  23. Varadhan, S.R.S.: Large deviations and applications. SIAM, Philadelphia 1984

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich 3 Mathematik, Technische Universität Berlin, Straße des 17. Juni, D-1000, Berlin 12

    E. Bolthausen

Authors
  1. E. Bolthausen
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Part of this work has been done when the author was at the Forschungsinstitut für Mathematik in Zürich. I would like to thank the members of the institute and especially Hans Föllmer for the kind hospitality

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bolthausen, E. Laplace approximations for sums of independent random vectors. Probab. Th. Rel. Fields 76, 167–206 (1987). https://doi.org/10.1007/BF00319983

Download citation

  • Received: 30 June 1986

  • Revised: 10 October 1986

  • Issue Date: October 1987

  • DOI: https://doi.org/10.1007/BF00319983

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

  • Entropy
  • Banach Space
  • Stochastic Process
  • Probability Theory
  • Statistical Theory
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