Mathematical Notes

, Volume 97, Issue 5–6, pp 878–891 | Cite as

On the asymptotic Laplace method and its application to random chaos

  • D. A. KorshunovEmail author
  • V. I. Piterbarg
  • E. Hashorva


The asymptotics of the multidimensional Laplace integral for the case in which the phase attains its minimum on an arbitrary smooth manifold is studied. Applications to the study of the asymptotics of the distribution of Gaussian and Weibullian random chaoses are considered.


Laplace asymptotic method Gaussian chaos Weibullian chaos Gelfand-Leray differential form random chaos 


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  1. 1.
    V. I. Arnol’d, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Mappings, Vol. 2: Monodromy and Asymptotic Behavior of Integrals (Nauka, Moscow, 1984; Birkhäuser, 1987).Google Scholar
  2. 2.
    E. Combet, Intégrales exponentielles. Développements asymptotiques. Propriétés Lagrangiennes, in Lecture Notes in Math. (Springer-Verlag, Berlin, 1982), Vol. 9zbMATHGoogle Scholar
  3. 3.
    C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Vol. I: Asymptotic Methods and Perturbation Theory (Springer-Verlag, New York, 1999).CrossRefGoogle Scholar
  4. 4.
    M. V. Fedoryuk, Asymptotics: Integrals and Series, in Reference Mathematical Library (Nauka, Moscow, 1987) [in Russian].Google Scholar
  5. 5.
    O. E. Trofimov and D. G. Frizen, “The coefficients of the asymptotic expansion of integrals by the Laplace method,” Avtometriya 2, 94 (1981).Google Scholar
  6. 6.
    R. Wong, Asymptotic Approximations of Integrals, in Computer Science and Scientific Computing (Academic Press, Boston, MA, 1989).Google Scholar
  7. 7.
    N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals (Holt, Rinehart and Winston, New York, 1975).zbMATHGoogle Scholar
  8. 8.
    W. Fulks and J. O. Sather, “Asymptotics. II. Laplace’s method for multiple integrals,” Pacific J. Math. 11, 185–192 (1961).zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    J. Wojdylo, “Computing the coefficients in Laplace’s method,” SIAM Rev. 48(1), 76–96 (2006).zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    J. L. López, P. Pagola, and E. Pérez Sinusá, “A simplification of Laplace’s method: applications to the Gamma function and Gauss hypergeometric function,” J. Approx. Theory 161, 280–291 (2009).zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    M. V. Fedoryuk, Saddle-Point Method (Nauka, Moscow, 1977) [in Russian].zbMATHGoogle Scholar
  12. 12.
    Ph. Barbe, Approximation of Integrals over Asymptotic Sets with Applications to Probability and Statistics, arXiv: math/0312132 (2003).Google Scholar
  13. 13.
    K. W. Breitung, Asymptotic Approximations for Probability Integrals, in Lecture Notes in Math. (Springer-Verlag, Berlin, 1994), Vol. 1592.zbMATHGoogle Scholar
  14. 14.
    N. Wiener, “The homogeneous chaos,” Amer. J. Math. 60(4), 897–936 (1938).MathSciNetCrossRefGoogle Scholar
  15. 15.
    Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Ed. by M. Abramowitz and I. Stegun (Dover Publications, New York, 1972; Nauka, Moscow, 1979).zbMATHGoogle Scholar
  16. 16.
    W.-D. Richter, “Generalized spherical and simplicial coordinates,” J. Math. Anal. Appl. 336(2), 1187–1202 (2007).zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    E. Hashorva, D. Korshunov, and V. I. Piterbarg, “Asymptotic expansion of Gaussian chaos via probabilistic approach,” Extremes (2015).Google Scholar
  18. 18.
    D. A. Korshunov, V. I. Piterbarg, and E. Hashorva, “On extremal behavior of Gaussian chaos,” Dokl. Ross. Akad. Nauk 452(5), 483–485 (2013) [Dokl. Math. 88 (2), 566–568 (2013)].MathSciNetGoogle Scholar
  19. 19.
    S. Foss, D. Korshunov, and S. Zachary, An Introduction to Heavy-Tailed and Subexponential Distributions, in Springer Ser. Oper. Res. Financ. Eng. (Springer, New York, 2011).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  • D. A. Korshunov
    • 1
    • 2
    Email author
  • V. I. Piterbarg
    • 3
  • E. Hashorva
    • 4
  1. 1.Sobolev Institute of MathematicsRussian Academy of SciencesNovosibirskRussia
  2. 2.Lancaster UniversityLancasterUK
  3. 3.Lomonosov Moscow State UniversityMoscowRussia
  4. 4.University of LausanneLausanneSwitzerland

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