An extension of Sanov's theorem: application to the Gibbs conditioning principle

A large deviation principle is proved for the empirical measures of independent identically distributed random variables with a topology based on functions having only some exponential moments. The rate function differs from the usual relative entropy: It involves linear forms which are no longer measures. Following D.W. Stroock and O. Zeitouni, the Gibbs Conditioning Principle (GCP) is then derived with the help of the previous result. Besides a rather direct proof, the main improvements with respect to already published GCP’s are the following: Convergence holds in situations where the underlying log-Laplace transform (the pressure) may not be steep and the constraints are built on energy functions admitting only some finite exponential moments. Basic techniques from Orlicz spaces theory appear to be a powerful tool. (1) Modal-X, Université Paris 10, Bâtiment G 200, Av. de la République, 92001 Nanterre cedex, France (2) CMAP, Ecole Polytechnique, 91128 Palaiseau cedex, France E-mail: Christian.Leonard@nospam.u-paris10.fr, najim@nospam.tsi.enst.fr Math. Subj. Class.: 60F10, 60G57 (primary); 46E30 (secondary)

[1]  E. Bolthausen,et al.  On the maximum entropy principle for uniformly ergodic Markov chains , 1989 .

[2]  Sandy L. Zabell,et al.  Large Deviations of the Sample Mean in General Vector Spaces , 1979 .

[3]  R. Rockafellar.  Integrals which are convex functionals. II , 1968 .

[4]  J. Doob.  Stochastic processes , 1953 .

[5]  Large deviations of U-empirical measures in strong topologies and applications☆ , 2002 .

[6]  S. Varadhan,et al.  Asymptotic evaluation of certain Markov process expectations for large time , 1975 .

[7]  M. Rao,et al.  Theory of Orlicz spaces , 1991 .

[8]  John R. Giles,et al.  Convex analysis with application in the differentiation of convex functions , 1982 .

[9]  Ofer Zeitouni,et al.  Microcanonical Distributions, Gibbs States, and the Equivalence of Ensembles , 1991 .

[10]  A. Acosta.  On large deviations of empirical measures in the τ-topology , 1994 .

[11]  I. Csiszár.  Sanov Property, Generalized $I$-Projection and a Conditional Limit Theorem , 1984 .

[12]  G. J. O. Jameson.  J. R. Giles, Convex analysis with application in the differentiation of convex functions (Research Notes in Mathematics No. 58, Pitman, 1982), £9.95. , 1983 .

[13]  The Gibbs principle for Markov jump processes , 1996 .

[14]  Piet Groeneboom,et al.  Large Deviation Theorems for Empirical Probability Measures , 1979 .

[15]  I. N. Sanov.  On the probability of large deviations of random variables , 1958 .

[16]  Christian Léonard.  Minimizers of Energy Functionals Under Not Very Integrable Constraints , 2000 .

[17]  Cramer's condition and Sanov's theorem , 1998 .

[18]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .