Conditional principles for random weighted measures

Nathael Gozlan

ESAIM: Probability and Statistics (2010)

  • Volume: 9, page 283-306
  • ISSN: 1292-8100

Abstract

top
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 , ((Zi)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.

How to cite

top

Gozlan, Nathael. "Conditional principles for random weighted measures." ESAIM: Probability and Statistics 9 (2010): 283-306. <http://eudml.org/doc/104338>.

@article{Gozlan2010,
abstract = { In this paper, we prove a conditional principle of Gibbs type for random weighted measures of the form $\{L_n=\frac\{1\}\{n\}\sum_\{i=1\}^nZ_i\delta_\{x_i^n\}\}$, ((Zi)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. },
author = {Gozlan, Nathael},
journal = {ESAIM: Probability and Statistics},
keywords = {Large deviations; transportation cost inequalities; conditional laws of large numbers; minimum entropy methods. ; minimum entropy methods},
language = {eng},
month = {3},
pages = {283-306},
publisher = {EDP Sciences},
title = {Conditional principles for random weighted measures},
url = {http://eudml.org/doc/104338},
volume = {9},
year = {2010},
}

TY - JOUR
AU - Gozlan, Nathael
TI - Conditional principles for random weighted measures
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 9
SP - 283
EP - 306
AB - In this paper, we prove a conditional principle of Gibbs type for random weighted measures of the form ${L_n=\frac{1}{n}\sum_{i=1}^nZ_i\delta_{x_i^n}}$, ((Zi)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.
LA - eng
KW - Large deviations; transportation cost inequalities; conditional laws of large numbers; minimum entropy methods. ; minimum entropy methods
UR - http://eudml.org/doc/104338
ER -

References

top
  1. S.G. Bobkov and F. Gotze, Exponential integrability and transportation cost related to logarithmic sobolev inequalities. J. Funct. Anal.163 (1999) 1–28.  
  2. J.M. Borwein and A.S. Lewis, Duality relationships for entropy-like minimization problems. SIAM J. Control Optim.29 (1991) 325–338.  
  3. J.M. Borwein and A.S. Lewis, Partially-finite programming in L1 and the exitence of maximum entropy estimates. SIAM J. Optim.3 (1993) 248–267.  
  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.  
  6. I. Csiszar, Sanov property, generalized I-projection and a conditional limit theorem. Ann. Prob.12 (1984) 768–793.  
  7. I. Csiszar, Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems. Ann. Statist.19 (1991) 2032–2066.  
  8. I. Csiszar, F. Gamboa and E. Gassiat, Mem pixel correlated solutions for generalized moment and interpolation problems. IEEE Trans. Inform. Theory45 (1999) 2253–2270.  
  9. D. Dacunha-Castelle and F. Gamboa, Maximum d'entropie et problèmes des moments. Ann. Inst. Henri Poincaré26 (1990) 567–596.  
  10. A. Dembo and O. Zeitouni, Large deviations techniques and applications. Second edition. Springer-Verlag (1998).  
  11. J.D. Deuschel and D.W. Stroock, Large deviations. Academic Press (1989).  
  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.  
  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.  
  15. F. Gamboa and E. Gassiat, Bayesian methods and maximum entropy for ill-posed inverse problems. Ann. Statist.25 (1997) 328–350.  
  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).  
  18. C. Léonard, Minimizer of energy functionals. Acta Math. Hungar.93 (2001) 281–325.  
  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).  
  23. R.T. Rockafellar and R. Wets, Variational Analysis. Springer-Verlag (1997).  
  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.  
  25. A. Van Der Vaart and J. Wellner, Weak convergence and empirical processes. Springer Series in Statistics. Springer (1995).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.