Probability Theory and Related Fields

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

Laplace approximations for sums of independent random vectors

  • E. Bolthausen
Article

Summary

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\({{\left( {n\phi \left( {\sum\limits_{i = 1}^n {{{X_i } \mathord{\left/ {\vphantom {{X_i } n}} \right. \kern-\nulldelimiterspace} n}} } \right)} \right)} \mathord{\left/ {\vphantom {{\left( {n\phi \left( {\sum\limits_{i = 1}^n {{{X_i } \mathord{\left/ {\vphantom {{X_i } n}} \right. \kern-\nulldelimiterspace} n}} } \right)} \right)} {Z_n }}} \right. \kern-\nulldelimiterspace} {Z_n }}\).

Keywords

Banach Space Stochastic Process Probability Theory Limit Theorem Random Vector 

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References

  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

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