×
Dedecker, Jérôme; Prieur, Clémentine

New dependence coefficients. Examples and applications to statistics. (English) Zbl 1061.62058

Probab. Theory Relat. Fields 132, No. 2, 203-236 (2005).
Summary: To measure the dependence between a real-valued random variable \(X\) and a \(\sigma\)-algebra \(\mathcal M\), we consider four distances between the conditional distribution function of \(X\) given \(\mathcal M\) and the distribution function of \(X\). The coefficients obtained are weaker than the corresponding mixing coefficients and may be computed in many situations. In particular, we show that they are well adapted to functions of mixing sequences, iterated random functions and dynamical systems.
Starting from a new covariance inequality, we study the mean integrated square error for estimating the unknown marginal density of a stationary sequence. We obtain optimal rates for kernel estimators as well as projection estimators on a well localized basis, under a minimal condition on the coefficients. Using recent results, we show that our coefficients may be also used to obtain various exponential inequalities, a concentration inequality for Lipschitz functions, and a Berry-Esseen type inequality.

Cited in 5 Reviews
Cited in 92 Documents

MSC:

62G07 Density estimation
60A10 Probabilistic measure theory
62G20 Asymptotic properties of nonparametric inference
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60E15 Inequalities; stochastic orderings
37N99 Applications of dynamical systems
PDFBibTeX XMLCite
\textit{J. Dedecker} and \textit{C. Prieur}, Probab. Theory Relat. Fields 132, No. 2, 203--236 (2005; Zbl 1061.62058)
Full Text: DOI

References:

[1] Andrews, J. Appl. Probab., 21, 930 (1984) · Zbl 0552.60049
[2] Barbour, Probab. Theory Relat. Fields, 116, 151 (2000) · Zbl 0976.37019
[3] Barron, Probab. Theory Relat. Fields, 113, 301 (1999) · Zbl 0946.62036 · doi:10.1007/s004400050210
[4] Berbee, H.C.P.: Random walks with stationary increments and renewal theory. Math. Cent. Tracts. Amsterdam, 1979 · Zbl 0443.60083
[5] Billingsley, P.: (1968) Convergence of probability measures. Wiley, New-York · Zbl 0172.21201
[6] Borovkova, Trans. Amer. Math. Soc., 353, 4261 (2001) · Zbl 0980.60031 · doi:10.1090/S0002-9947-01-02819-7
[7] Bradley, Michigan Math. J., 30, 69 (1983) · Zbl 0531.60033 · doi:10.1307/mmj/1029002789
[8] Bradley, R.C.: Introduction to Strong Mixing Conditions, Volume 1. Technical Report, Department of Mathematics, I. U. Bloomington, 2002
[9] Broise, Transformations dilatantes de l’intervalle et théorèmes limites. Études spectrales d’opérateurs de transfert et applications. Astérisque, 238, 1 (1996) · Zbl 0988.37032
[10] Castaing, C., Raynaud de Fitte, P., Valadier, M.: Young Measures on Topological Spaces. With Applications in Control Theory and Probability Theory. Kluwer Academic Publishers, Dordrecht, 2004 · Zbl 1067.28001
[11] Collet, Probab. Theory. Relat. Fields, 123, 301 (2002) · Zbl 1087.37028 · doi:10.1007/s004400200204
[12] Deddens, J., Peligrad, M., Yang, T.: (1987) On strong consistency of kernel estimators under dependence assumptions. Mathematical statistics and probability theory Vol. B 33-41, (Bad Tatzmannsdorf, 1986) Reidel, Dordrecht · Zbl 0631.62038
[13] Dedecker, A new covariance inequality and applications. Stochastic Process. Appl., 106, 63 (2003) · Zbl 1075.60513
[14] Dedecker, J., Prieur, C.: (2003) Coupling for τ-dependent sequences and applications. Preprint LSP 2003-02. Université Paul Sabatier, Toulouse. Accepted for publication in J. Theoret. Probab. · Zbl 1067.60008
[15] Dedecker, C. R. Acad. Sci. Paris, Ser I, 338, 805 (2004) · Zbl 1044.60002
[16] Dedecker, Ann. Inst. H. Poincaré Probab. Statist., 36, 1 (2000) · Zbl 0949.60049 · doi:10.1016/S0246-0203(00)00111-4
[17] Dehling, Z. Wahrsch. Verw. Gebiete, 62, 39 (1983) · Zbl 0479.60013 · doi:10.1007/BF00532161
[18] Delyon, B.: (1990) Limit theorems for mixing processes. Tech. Report 546 IRISA, Rennes I
[19] DeVore, R.A., Lorentz, G.G.: Constructive approximation. Springer, Berlin, Heidelberg, New York, 1993 · Zbl 0797.41016
[20] Diaconis, SIAM Rev., 41, 45 (1999) · Zbl 0926.60056
[21] Dobrushin, Theory Probab. Appl., 15, 458 (1970) · Zbl 0264.60037 · doi:10.1137/1115049
[22] Doukhan, P.: Mixing: properties and examples. Lecture Notes in Statist. 85, Springer-Verlag, 1994 · Zbl 0801.60027
[23] Doukhan, Stochastic Process. Appl., 84, 313 (1999) · Zbl 0996.60020 · doi:10.1016/S0304-4149(99)00055-1
[24] Dudley, R.M.: Real analysis and probability. Wadworsth Inc., Belmont, California, 1989 · Zbl 0686.60001
[25] Hofbauer, Math. Z., 180, 119 (1982) · Zbl 0485.28016 · doi:10.1007/BF01215004
[26] Ibragimov, Theory Probab. Appl., 7, 349 (1962) · Zbl 0119.14204 · doi:10.1137/1107036
[27] Lasota, Trans. Amer. Math. Soc., 186, 481 (1974) · Zbl 0298.28015
[28] Ledoux, M., Talagrand, M.: Probability in Banach spaces. Springer, New-York, 1991 · Zbl 0748.60004
[29] Major, J. Multivariate Anal., 8, 487 (1978) · Zbl 0408.60028 · doi:10.1016/0047-259X(78)90029-5
[30] Merlevède, F., Peligrad, M.: (2002) On the coupling of dependent random variables and applications. Empirical process techniques for dependent data. 171-193 Birkhäuser · Zbl 1038.60021
[31] Morita, J. Math. Soc. Japan, 46, 309 (1994) · Zbl 0824.60017
[32] Peligrad, Stat. Prob. Letters, 60, 201 (2002) · Zbl 1020.60022 · doi:10.1016/S0167-7152(02)00318-8
[33] Philipp, W., Stout, W.: Almost sure invariance principles for partial sums of weakly dependent random variables. Mem. Amer. Math. Soc 161, (1975) · Zbl 0361.60007
[34] Prieur, ESAIM, Probab. & Statist., 5, 51 (2001) · Zbl 1054.60030
[35] Rachev, S.T., Rüschendorf, L.: Mass Transportation Problems, Volume 1: Theory. Springer. Berlin, Heidelberg, New-York, 1998 · Zbl 0990.60500
[36] Rio, Ann. Probab., 23, 1188 (1995)
[37] Rio, Probab. Theory Relat. Fields, 104, 255 (1996) · Zbl 0838.60017
[38] Rio, E.: Théorie asymptotique des processus aléatoires faiblement dépendants. Collection Mathématiques & Applications 31, Springer, Berlin, (2000a) · Zbl 0944.60008
[39] Rio, E.: Inégalités de Hoeffding pour les fonctions lipschitziennes de suites dépendantes. C. R. Acad. Sci. Paris Série I 330, 905-908 (2000b) · Zbl 0961.60032
[40] Rosenblatt, A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. U. S. A., 42, 43 (1956) · Zbl 0070.13804
[41] Rozanov, Theory Probab. Appl., 4, 178 (1959) · Zbl 0092.33502 · doi:10.1137/1104015
[42] van der Vaart, A.W., Wellner, J.A.: Weak convergence and empirical processes. Springer, Berlin, 1996 · Zbl 0862.60002
[43] Viennet, Probab. Theory Relat. Fields, 107, 467 (1997) · Zbl 0933.62029 · doi:10.1007/s004400050094
[44] Yu, Probab. Theory Relat. Fields, 95, 357 (1993) · Zbl 0792.60018 · doi:10.1007/BF01192169
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.