The Annals of Probability

Asymptotic expansions for the Laplace approximations of sums of Banach space-valued random variables

Sergio Albeverio and Song Liang

Full-text: Open access

Abstract

Let Xi, iN, be i.i.d. B-valued random variables, where B is a real separable Banach space. Let Φ be a smooth enough mapping from B into R. An asymptotic evaluation of Zn=E(exp(nΦ(∑i=1nXi/n))), up to a factor (1+o(1)), has been gotten in Bolthausen [Probab. Theory Related Fields 72 (1986) 305–318] and Kusuoka and Liang [Probab. Theory Related Fields 116 (2000) 221–238]. In this paper, a detailed asymptotic expansion of Zn as n→∞ is given, valid to all orders, and with control on remainders. The results are new even in finite dimensions.

Article information

Source
Ann. Probab., Volume 33, Number 1 (2005), 300-336.

Dates
First available in Project Euclid: 11 February 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1108141728

Digital Object Identifier
doi:10.1214/009117904000001017

Mathematical Reviews number (MathSciNet)
MR2118867

Zentralblatt MATH identifier
1092.62024

Subjects
Primary: 62E20: Asymptotic distribution theory 60F10: Large deviations 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)

Keywords
Laplace approximation asymptotic expansions i.i.d. random vectors Banach space-valued random variables

Citation

Albeverio, Sergio; Liang, Song. Asymptotic expansions for the Laplace approximations of sums of Banach space-valued random variables. Ann. Probab. 33 (2005), no. 1, 300--336. doi:10.1214/009117904000001017. https://projecteuclid.org/euclid.aop/1108141728


Export citation

References

  • Albeverio, S. (1997). Wiener and Feynman---path integrals and their applications. In Proceedings of the Symposium in Applied Mathematics 52 163--194. Amer. Math. Soc., Providence, RI.
  • Albeverio, S., Röckle, H. and Steblovskaya, V. (2000). Asymptotic for Ornstein--Uhlenbeck semigroups perturbed by potentials over Banach spaces. Stochastics Stochastics Rep. 69 195--238.
  • Albeverio, S. and Steblovskaya, V. (1999). Asymptotics of infinite-dimensional integrals with respect to smooth measures. I. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2 529--556.
  • Araujo, A. and Giné, E. (1980). The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley, New York.
  • Arnold, V. I., Guseĭ n-Zade, S. M. and Varchenko, A. N. (1988). Singularities of Differentiable Maps, I, II. Birkhäuser, Boston.
  • Ben Arous, G. (1988). The Laplace and stationary phase methods on Wiener space. Stochastics 25 125--153.
  • Benkus, V. and Götze, F. (1997). Uniform rates of convergence in the CLT for quadratic forms in multidimensional spaces. Probab. Theory Related Fields 109 367--416.
  • Bolthausen, E. (1986). Laplace approximations for sums of independent random vectors. Probab. Theory Related Fields 72 305--318.
  • Bolthausen, E. (1987). Laplace approximations for sums of independent random vectors. II. Degenerate maxima and manifolds of maxima. Probab. Theory Related Fields 76 167--206.
  • de Acosta, A. (1992). Moderate deviations and associated Laplace approximations for sums of independent random vectors. Trans. Amer. Math. Soc. 329 357--375.
  • Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York.
  • Donsker, M. D. and Varadhan, S. R. S. (1975). Asymptotic evaluation of certain Markov process expectations for large time---I. Comm. Pure Appl. Math. 27 1--47.
  • Donsker, M. D. and Varadhan, S. R. S. (1976). Asymptotic evaluation of certain Markov process expectations for large time---III. Comm. Pure Appl. Math. 29 389--461.
  • Ellis, A. J. (1978). Topics in Banach Spaces of Continuous Functions. Macmillan, New Delhi.
  • Ellis, A. J. (1985). Entropy, Large Deviations, and Statistical Mechanics. Springer, New York.
  • Ellis, A. J. and Rosen, J. S. (1981). Asymptotic analysis of Gaussian integrals II: Manifold of mimimum points. Commun. Math. Phys. 82 153--181.
  • Ellis, A. J. and Rosen, J. S. (1982). Asymptotic analysis of Gaussian integrals I: Isolated minimum points. Trans. Amer. Math. Soc. 273 447--481.
  • Gnedenko, B. V. and Kolmogorov, A. N. (1968). Limit Distributions for Sums of Independent Random Variables. Addison--Wesley, Reading, MA.
  • Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
  • Kuo, H.-H. (1975). Gaussian Measures in Banach Spaces. Springer, Berlin.
  • Kusuoka, S. and Liang, S. (2000). Laplace approximations for sums of independent random vectors. Probab. Theory Related Fields 116 221--238.
  • Kusuoka, S. and Liang, S. (2001). Laplace approximations for diffusion processes on torus: Nondegenerate case. J. Math. Sci. Univ. Tokyo 8 43--70.
  • Kusuoka, S. and Tamura, Y. (1984). Gibbs measures for mean field potentials. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 223--245.
  • Liang, S. (2000). Laplace approximations for sums of independent random vectors---the degenerate case. J. Math. Sci. Univ. Tokyo 7 195--220.
  • Liang, S. (2003). Laplace approximations of the large deviations for diffusion processes on Euclidean spaces. Unpublished manuscript.
  • Lu, Y. C. (1976). Singularity Theory and an Introduction to Catastrophe Theory. Springer, New York.
  • Piterbarg, V. I. and Fatalov, V. R. (1995). The Laplace method for probability measures in Banach spaces. Uspekhi Mat. Nauk 50 57--150. (In Russian.) Translation in Russian Math. Surveys 50 1151--1239.