The empirical distribution function for dependent variables : asymptotic and nonasymptotic results in 𝕃 p
ESAIM: Probability and Statistics, Tome 11 (2007), pp. 102-114.

Considering the centered empirical distribution function F n -F as a variable in 𝕃 p (μ), we derive non asymptotic upper bounds for the deviation of the 𝕃 p (μ)-norms of F n -F as well as central limit theorems for the empirical process indexed by the elements of generalized Sobolev balls. These results are valid for a large class of dependent sequences, including non-mixing processes and some dynamical systems.

DOI : 10.1051/ps:2007009
Classification : 60F10, 62G30
Mots clés : deviation inequalities, weak dependence, Cramér-von Mises statistics, empirical process, expanding maps
@article{PS_2007__11__102_0,
     author = {Dedecker, J\'er\^ome and Merlev\`ede, Florence},
     title = {The empirical distribution function for dependent variables : asymptotic and nonasymptotic results in ${\mathbb {L}}^p$},
     journal = {ESAIM: Probability and Statistics},
     pages = {102--114},
     publisher = {EDP-Sciences},
     volume = {11},
     year = {2007},
     doi = {10.1051/ps:2007009},
     mrnumber = {2299650},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps:2007009/}
}
TY  - JOUR
AU  - Dedecker, Jérôme
AU  - Merlevède, Florence
TI  - The empirical distribution function for dependent variables : asymptotic and nonasymptotic results in ${\mathbb {L}}^p$
JO  - ESAIM: Probability and Statistics
PY  - 2007
SP  - 102
EP  - 114
VL  - 11
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps:2007009/
DO  - 10.1051/ps:2007009
LA  - en
ID  - PS_2007__11__102_0
ER  - 
%0 Journal Article
%A Dedecker, Jérôme
%A Merlevède, Florence
%T The empirical distribution function for dependent variables : asymptotic and nonasymptotic results in ${\mathbb {L}}^p$
%J ESAIM: Probability and Statistics
%D 2007
%P 102-114
%V 11
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps:2007009/
%R 10.1051/ps:2007009
%G en
%F PS_2007__11__102_0
Dedecker, Jérôme; Merlevède, Florence. The empirical distribution function for dependent variables : asymptotic and nonasymptotic results in ${\mathbb {L}}^p$. ESAIM: Probability and Statistics, Tome 11 (2007), pp. 102-114. doi : 10.1051/ps:2007009. http://archive.numdam.org/articles/10.1051/ps:2007009/

[1] K. Azuma, Weighted sums of certain dependent random variables. Tôkohu Math. J. 19 (1967) 357-367. | MR | Zbl

[2] H.C.P. Berbee, Random walks with stationary increments and renewal theory. Mathematical Centre Tracts 112, Mathematisch Centrum, Amsterdam (1979). | MR | Zbl

[3] L. Birgé and P. Massart, An adaptive compression algorithm in Besov Spaces. Constr. Approx. 16 (2000) 1-36. | MR | Zbl

[4] P. Collet, S. Martinez and B. Schmitt, Exponential inequalities for dynamical measures of expanding maps of the interval. Probab. Theory Relat. Fields 123 (2002) 301-322. | MR | Zbl

[5] J. Dedecker and F. Merlevède, The conditional central limit theorem in Hilbert spaces. Stoch. Processes Appl. 108 (2003) 229-262. | MR | Zbl

[6] J. Dedecker and C. Prieur, Coupling for τ-dependent sequences and applications. J. Theoret. Probab. 17 (2004) 861-885. | MR | Zbl

[7] J. Dedecker and C. Prieur, New dependence coefficients. Examples and applications to statistics. Probab. Theory Relat. Fields 132 (2005) 203-236. | MR | Zbl

[8] J. Dedecker and E. Rio, On the functional central limit theorem for stationary processes. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) 1-34. | EuDML | Numdam | MR | Zbl

[9] P. Doukhan, P. Massart and E. Rio, Invariance principle for absolutely regular empirical processes. Ann. Inst. H. Poincaré Probab. Statist. 31 (1995) 393-427. | EuDML | Numdam | MR | Zbl

[10] M.I. Gordin, The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188 (1969) 739-741. | Zbl

[11] P. Massart, The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. Ann. Probab. 18 (1990) 1269-1283. | Zbl

[12] F. Merlevède and M. Peligrad, On the coupling of dependent random variables and applications, in Empirical process techniques for dependent data, Birkhäuser (2002) 171-193. | Zbl

[13] P. Oliveira and C. Suquet, 𝕃 2 ([0,1]) weak convergence of the empirical process for dependent variables, in Wavelets and statistics (Villard de Lans 1994), Lect. Notes Statist. 103 (1995) 331-344. | Zbl

[14] P. Oliveira and C. Suquet, Weak convergence in 𝕃 p ([0,1]) of the uniform empirical process under dependence. Statist. Probab. Lett. 39 (1998) 363-370. | Zbl

[15] I.F. Pinelis, An approach to inequalities for the distributions of infinite-dimensional martingales, in Probability in Banach spaces, Proc. Eight Internat. Conf. 8 (1992) 128-134. | Zbl

[16] E. Rio, Inégalités de Hoeffding pour les fonctions lipschitziennes de suites dépendantes. C. R. Acad. Sci. Paris Série I 330 (2000) 905-908. | Zbl

[17] A.W. Van Der Vaart, Bracketing smooth functions. Stoch. Processes Appl. 52 (1994) 93-105. | Zbl

[18] W.A. Woyczyński, A central limit theorem for martingales in Banach spaces. Bull. Acad. Polon. Sci. Sr. Sci. Math. Astronom. Phys. 23 (1975) 917-920. | Zbl

[19] V.V. Yurinskii, Exponential bounds for large deviations. Theory Prob. Appl. 19 (1974) 154-155. | Zbl

Cité par Sources :