Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-22T20:40:29.487Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability of precise Laplace's method under approximations;Applications

Published online by Cambridge University Press:  15 August 2002

A. Guionnet
A. Guionnet*
Affiliation:
URA 743 du CNRS, bâtiment 425, Université de Paris Sud, 91405 Orsay, France.
Get access

Abstract

We study the fluctuations around non degenerate attractorsof the empirical measure under mean field Gibbs measures.We prove that a mild change of the densitiesof these measures does not affect the central limit theorems.We apply this result to generalize the assumptionsof [3] and [12] on the densities of the Gibbs measures toget precise Laplace estimates.

Keywords

Large deviationcentral limit theoremU-statistics.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 3 , 1999 , pp. 67 - 88
Copyright
© EDP Sciences, SMAI, 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arcones, M.A. and Gine, E., Limit Theorems for U-processes. Ann. Probab. 21 (1993) 1494-1542. CrossRef
Arcones, M.A. and Gine, E., On the bootstrap of U and V statistics. Ann. Stat. 20 (1992) 655-674. CrossRef
Ben Arous, G. and Brunaud, M., Méthode de Laplace : Étude variationnelle des fluctuations de diffusions de type "champ moyen''. Stochastics 31-32 (1990) 79-144.
M.Sh. Birman, A proof of the Fredholm trace formula as an application of a simple embedding for kernels of integral operators of trace class in $L^2(\mathbb{R}^m)$ . Lith-Mat-R-89-30 (1989).
E. Bolthausen, Laplace approximation for sums of independent random vectors I. Prob. Th. Rel. Fields 72 (1986) 305-318.
C. Borell, On the integrability of Banach space valued Walsh polynomials. Séminaire de probabilités XIII. Lecture Notes in Math. 721 (1979) 1-3.
Dawson, D.A., Critical dynamics and fluctuations for a mean field model of cooperative behavior. J. Stat. Phys. 31 (1983) 247-308. CrossRef
De Acosta, A. and Gine, E., Convergence of moments and related Functionals in the central limit Theorem in Banach spaces. Z. Wahrsch. Verw. Gebiete 48 (1979) 213-231. CrossRef
A. Dembo and O. Zeitouni, Large deviations techniques and Applications. Jones and Bartlett (1992).
J.D. Deuschel and D.W. Stroock, Large deviations. Academic press (1989).
Hitsuda, M. and Tanaka, H., Central limit Theorem for a simple diffusion model for interacting particles. Hiroshima Math. J. 11 (1981) 415-423.
S. Kusuoka and Y. Tamura, Gibbs measures for mean field potentials. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1984).
Mac Kean, H.P., Fluctuations in the kinetic theory of gases. Comm. Pure Appl. Math. 28 (1975) 435-455. CrossRef
K.R. Parthasarathy, Probability measures on Metric Spaces. Academic Press Inc., New York (1968).
V.H. De La Pena, Decoupling and Khintchine's inequalities for U-statistics. Ann. Probab. 20 (1992) 1877-1892.
W. Rudin, Real and complex analysis, second edition, Springer.
Shiga, T. and Tanaka, H., Central limit Theorem for a system of markovian particles with mean field interaction. Z. Wahrsch. Verw. Gebiete 69 (1985) 439-459. CrossRef
B. Simon, Trace ideals and their applications. London Mathematical Society Lecture Notes series 35, Cambridge University press (1977).
A.-S. Sznitman, Non linear reflecting diffusion process and the propagation of chaos and fluctuations associated.
Tanaka, H., Limit Theorems for certain diffusion processes. in Proc. of the Taniguchi Symp, Katata (1982) 469-488, Tokyo, Kinokuniya (1984). J. Funct. Anal. 56 (1984) 311-336.