Probability Theory and Related Fields

, Volume 72, Issue 2, pp 305–318 | Cite as

Laplace approximations for sums of independent random vectors

  • E. Bolthausen


LetXi,iɛN, be i.i.d.B-valued random variables whereB is a real separable Banach space, and Φ a mappingBR. Under some conditions an asymptotic evaluation of\(Z_n = E\left( {\exp \left( {n\phi \left( {\sum\limits_{i = 1}^n {{{X_i } \mathord{\left/ {\vphantom {{X_i } n}} \right. \kern-\nulldelimiterspace} n}} } \right)} \right)} \right)\) is possible, up to a factor (1+o(1)). This also leads to a limit theorem for the appropriately normalized sums\(\sum\limits_{i = 1}^n {X_i } \) under the law transformed by the density exp\({{\left( {n\phi \left( {\sum\limits_{i = 1}^n {{{X_i } \mathord{\left/ {\vphantom {{X_i } n}} \right. \kern-\nulldelimiterspace} n}} } \right)} \right)} \mathord{\left/ {\vphantom {{\left( {n\phi \left( {\sum\limits_{i = 1}^n {{{X_i } \mathord{\left/ {\vphantom {{X_i } n}} \right. \kern-\nulldelimiterspace} n}} } \right)} \right)} {Z_n }}} \right. \kern-\nulldelimiterspace} {Z_n }}\).


Banach Space Stochastic Process Probability Theory Limit Theorem Random Vector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    de Acosta, A., Giné, E.: 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
  2. 2.
    Azencott, R.: Grandes déviations et applications. Saint-Flour VIII 1978. Lecture Notes in Math.774. Berlin-Heidelberg-New York: Springer 1980Google Scholar
  3. 3.
    Borell, C.: Gaussian Radon measures on locally convex spaces. Math. Scand.38, 265–284 (1976)Google Scholar
  4. 4.
    Csziszar, I.:I-divergence geometry of probability distributions and minimization problems. Ann. Probab.3, 146–158 (1975)Google Scholar
  5. 5.
    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
  6. 6.
    Ellis, R.S., Rosen, J.S.: Asymptotic analysis of Gaussian integrals I, II. Trans. Am. Math. Soc.273, 447–481 and Comm. Math. Phys.82, 153–181 (1982)Google Scholar
  7. 7.
    Hoffmann-Jørgensen, J.: Probability in Banach spaces, Saint-Flour VI-1976, Lecture Notes in Math.598. Berlin-Heidelberg-New York: Springer 1977Google Scholar
  8. 8.
    Jain, N.C.: Central limit theorem in Banach spaces, Proc. First Conf. on Prob. in Banach Spaces Oberwolfach 1975. Lecture Notes Math.526. Berlin-Heidelberg-New York: Springer 1976Google Scholar
  9. 9.
    Kuo, H.-H.: Gaussian measures in Banach spaces. Lecture Notes Math.463. Berlin-Heidelberg-New York: Springer 1975Google Scholar
  10. 10.
    Kusuoka, Sh., Tamura, Y.: The convergence of Gibbs measures associated with mean field potentials. J. Fac. Sci., Univ. Tokyo, Sect. 1A31, 223–245 (1984)Google Scholar
  11. 11.
    Martin-Löf, A.: Laplace approximation for sums of independent random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb.59, 101–115 (1982)Google Scholar
  12. 12.
    Pincus, M.: Gaussian processes and Hammerstein integral equations. Trans. Am. Math. Soc.134, 193–216 (1968)Google Scholar
  13. 13.
    Schilder, M.: Some asymptotic formulae for Wiener integrals. Trans. Am. Math. Soc.125, 63–85 (1966)Google Scholar
  14. 14.
    Varadhan, S.R.S.: Asymptotic probabilities and differential equations. Comm. Pure Appl. Math.19, 261–286 (1966)Google Scholar
  15. 15.
    Yurinskii, V.V.: Exponential inequalities for sums of random vectors. J. Multivariate Anal.6, 473–499 (1966)Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • E. Bolthausen
    • 1
  1. 1.Fachbereich MathematikTechnische Universität BerlinBerlin 12

Personalised recommendations