Estimation of the density of a determinantal process
Confluentes Mathematici, Tome 5 (2013) no. 1, pp. 3-23.

We consider the problem of estimating the density Π of a determinantal process N from the observation of n independent copies of it. We use an aggregation procedure based on robust testing to build our estimator. We establish non-asymptotic risk bounds with respect to the Hellinger loss and deduce, when n goes to infinity, uniform rates of convergence over classes of densities Π of interest.

Reçu le : 2012-04-08
Accepté le : 2013-03-13
Accepté après révision le : 2013-06-01
Publié le : 2017-03-26
Classification : 62G07,  62M30
Mots clés: Determinantal process - Density estimation- Oracle inequality - Hellinger distance
     author = {Yannick Baraud},
     title = {Estimation of the density of a determinantal process},
     journal = {Confluentes Mathematici},
     publisher = {Institut Camille Jordan},
     volume = {5},
     number = {1},
     year = {2013},
     pages = {3-23},
     doi = {10.5802/cml.1},
     language = {en},
     url = {}
Yannick Baraud. Estimation of the density of a determinantal process. Confluentes Mathematici, Tome 5 (2013) no. 1, pp. 3-23. doi : 10.5802/cml.1.

[1] N. Akakpo Estimation adaptative par selection de partitions en rectangles dyadiques (2009) (Ph. D. Thesis)

[2] Greg W. Anderson; Alice Guionnet; Ofer Zeitouni An introduction to random matrices, Cambridge Studies in Advanced Mathematics, Volume 118, Cambridge University Press, Cambridge, 2010, xiv+492 pages

[3] Jinho Baik; Eric M. Rains Algebraic aspects of increasing subsequences, Duke Math. J., Volume 109 (2001) no. 1, pp. 1-65 | Article

[4] Yannick Baraud Estimator selection with respect to Hellinger-type risks, Probab. Theory Related Fields, Volume 151 (2011) no. 1-2, pp. 353-401 | Article

[5] Lucien Birgé Model selection via testing: an alternative to (penalized) maximum likelihood estimators, Ann. Inst. H. Poincaré Probab. Statist., Volume 42 (2006) no. 3, pp. 273-325 | Article

[6] Alexei Borodin; Persi Diaconis; Jason Fulman On adding a list of numbers (and other one-dependent determinantal processes), Bull. Amer. Math. Soc. (N.S.), Volume 47 (2010) no. 4, pp. 639-670 | Article

[7] N. G. de Bruijn On some multiple integrals involving determinants, J. Indian Math. Soc. (N.S.), Volume 19 (1955), p. 133-151 (1956)

[8] Reinhard Hochmuth Wavelet characterizations for anisotropic Besov spaces, Appl. Comput. Harmon. Anal., Volume 12 (2002) no. 2, pp. 179-208 | Article

[9] J. Ben Hough; Manjunath Krishnapur; Yuval Peres; Bálint Virág Determinantal processes and independence, Probab. Surv., Volume 3 (2006), pp. 206-229 | Article

[10] J. Ben Hough; Manjunath Krishnapur; Yuval Peres; Bálint Virág Zeros of Gaussian analytic functions and determinantal point processes, University Lecture Series, Volume 51, American Mathematical Society, Providence, RI, 2009, x+154 pages

[11] Russell Lyons Determinantal probability measures, Publ. Math. Inst. Hautes Études Sci. (2003) no. 98, pp. 167-212 | Article

[12] Saunders Mac Lane; Garrett Birkhoff Algebra, Chelsea Publishing Co., New York, 1988, xx+626 pages