Conditional principles for random weighted measures
ESAIM: Probability and Statistics, Tome 9 (2005), pp. 283-306.

In this paper, we prove a conditional principle of Gibbs type for random weighted measures of the form L n =1 n i=1 n Z i δ x i n , (Z i ) i being a sequence of i.i.d. real random variables. Our work extends the preceding results of Gamboa and Gassiat (1997), in allowing to consider thin constraints. Transportation-like ideas are used in the proof.

DOI : https://doi.org/10.1051/ps:2005016
Classification : 60E15,  60F10
Mots clés : large deviations, transportation cost inequalities, conditional laws of large numbers, minimum entropy methods
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     author = {Gozlan, Nathael},
     title = {Conditional principles for random weighted measures},
     journal = {ESAIM: Probability and Statistics},
     pages = {283--306},
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Gozlan, Nathael. Conditional principles for random weighted measures. ESAIM: Probability and Statistics, Tome 9 (2005), pp. 283-306. doi : 10.1051/ps:2005016. http://www.numdam.org/articles/10.1051/ps:2005016/

[1] S.G. Bobkov and F. Gotze, Exponential integrability and transportation cost related to logarithmic sobolev inequalities. J. Funct. Anal. 163 (1999) 1-28. | Zbl 0924.46027

[2] J.M. Borwein and A.S. Lewis, Duality relationships for entropy-like minimization problems. SIAM J. Control Optim. 29 (1991) 325-338. | Zbl 0797.49030

[3] J.M. Borwein and A.S. Lewis, Partially-finite programming in L 1 and the exitence of maximum entropy estimates. SIAM J. Optim. 3 (1993) 248-267. | Zbl 0780.49015

[4] P. Cattiaux and N. Gozlan, Deviations lower bounds and conditional principles. Prépublications de l'Université Paris 10, Nanterre (2002).

[5] I. Csiszar, I-divergence geometry of probability distributions and minimization problems. Ann. Prob. 3 (1975) 146-158. | Zbl 0318.60013

[6] I. Csiszar, Sanov property, generalized I-projection and a conditional limit theorem. Ann. Prob. 12 (1984) 768-793. | Zbl 0544.60011

[7] I. Csiszar, Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems. Ann. Statist. 19 (1991) 2032-2066. | Zbl 0753.62003

[8] I. Csiszar, F. Gamboa and E. Gassiat, Mem pixel correlated solutions for generalized moment and interpolation problems. IEEE Trans. Inform. Theory 45 (1999) 2253-2270. | Zbl 0958.94002

[9] D. Dacunha-Castelle and F. Gamboa, Maximum d'entropie et problèmes des moments. Ann. Inst. Henri Poincaré 26 (1990) 567-596. | Numdam | Zbl 0788.62007

[10] A. Dembo and O. Zeitouni, Large deviations techniques and applications. Second edition. Springer-Verlag (1998). | MR 1619036 | Zbl 0896.60013

[11] J.D. Deuschel and D.W. Stroock, Large deviations. Academic Press (1989). | MR 997938 | Zbl 0705.60029

[12] R.S. Ellis, J. Gough and J.V. Pulé, The large deviation principle for measures with random weights. Rev. Math. Phys. 5 (1993) 659-692. | Zbl 0797.60027

[13] F. Gamboa, Méthode du maximum d'entropie sur la moyenne et applications. Thèse Orsay (1989).

[14] F. Gamboa and E. Gassiat, Maximum d'entropie et problèmes des moments: Cas multidimensionnel. Probab. Math. Statist. 12 (1991) 67-83. | Zbl 0766.60003

[15] F. Gamboa and E. Gassiat, Bayesian methods and maximum entropy for ill-posed inverse problems. Ann. Statist. 25 (1997) 328-350. | Zbl 0871.62010

[16] N. Gozlan, Principe conditionnel de Gibbs pour des contraintes fines approchées et inégalités de transport. Université Paris 10-Nanterre (2005).

[17] J.B. Hirriart-Urruty and C. Lemaréchal, Fundamentals of convex analysis. Springer-Verlag (2001). | MR 1865628 | Zbl 0998.49001

[18] C. Léonard, Minimizer of energy functionals. Acta Math. Hungar. 93 (2001) 281-325. | Zbl 1002.49017

[19] C. Léonard, A convex optimization problem arising from probabilistic questions. Prépublications de l'Université Paris 10-Nanterre (2004).

[20] C. Léonard, Dominating points and entropic projections. Prépublications de l'Université Paris 10-Nanterre (2004).

[21] P. Massart, Saint-Flour Lecture Notes (2003).

[22] J. Najim, A Cramer type theorem for weighted random variables. Electronic J. Probab. 7 (2002). | MR 1887624 | Zbl 1011.60005

[23] R.T. Rockafellar and R. Wets, Variational Analysis. Springer-Verlag (1997). | MR 1491362 | Zbl 0888.49001

[24] D.W. Stroock and O. Zeitouni, Microcanonical distributions, Gibbs states and the equivalence of ensembles, R. Durret and H. Kesten Eds., Birkhäuser. Festschrift in honour of F. Spitzer (1991) 399-424. | Zbl 0745.60105

[25] A. Van Der Vaart and J. Wellner, Weak convergence and empirical processes. Springer Series in Statistics. Springer (1995). | MR 1385671 | Zbl 0862.60002

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