In this paper, a very useful lemma (in two versions) is proved: it simplifies notably the essential step to establish a Lindeberg central limit theorem for dependent processes. Then, applying this lemma to weakly dependent processes introduced in Doukhan and Louhichi (1999), a new central limit theorem is obtained for sample mean or kernel density estimator. Moreover, by using the subsampling, extensions under weaker assumptions of these central limit theorems are provided. All the usual causal or non causal time series: gaussian, associated, linear, ARCH(), bilinear, Volterra processes, , enter this frame.
Mots-clés : central limit theorem, Lindeberg method, weak dependence, kernel density estimation, subsampling
@article{PS_2008__12__154_0,
author = {Bardet, Jean-Marc and Doukhan, Paul and Lang, Gabriel and Ragache, Nicolas},
title = {Dependent {Lindeberg} central limit theorem and some applications},
journal = {ESAIM: Probability and Statistics},
pages = {154--172},
publisher = {EDP-Sciences},
volume = {12},
year = {2008},
doi = {10.1051/ps:2007053},
mrnumber = {2374636},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2007053/}
}
TY - JOUR AU - Bardet, Jean-Marc AU - Doukhan, Paul AU - Lang, Gabriel AU - Ragache, Nicolas TI - Dependent Lindeberg central limit theorem and some applications JO - ESAIM: Probability and Statistics PY - 2008 SP - 154 EP - 172 VL - 12 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2007053/ DO - 10.1051/ps:2007053 LA - en ID - PS_2008__12__154_0 ER -
%0 Journal Article %A Bardet, Jean-Marc %A Doukhan, Paul %A Lang, Gabriel %A Ragache, Nicolas %T Dependent Lindeberg central limit theorem and some applications %J ESAIM: Probability and Statistics %D 2008 %P 154-172 %V 12 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2007053/ %R 10.1051/ps:2007053 %G en %F PS_2008__12__154_0
Bardet, Jean-Marc; Doukhan, Paul; Lang, Gabriel; Ragache, Nicolas. Dependent Lindeberg central limit theorem and some applications. ESAIM: Probability and Statistics, Tome 12 (2008), pp. 154-172. doi : 10.1051/ps:2007053. http://www.numdam.org/articles/10.1051/ps:2007053/
[1] , Non strong mixing autoregressive processes. J. Appl. Probab. 21 (1984) 930-934. | MR | Zbl
[2] , Convergence of Probability Measures. Wiley, New-York (1968). | MR | Zbl
[3] and , Rates in the central limit theorem for weakly dependent random variables. J. Math. Sci. 122 (2004) 3343-3358. | MR | Zbl
[4] and , Strong Invariance Principle for Dependent Multi-indexed Random Variables. Doklady Mathematics 72 (2005) 503-506. | MR | Zbl
[5] and , A triangular central limit theorem under a new weak dependence condition. Stat. Prob. Letters 47 (2000) 61-68. | MR | Zbl
[6] , Mixing: Properties and Examples. Lect. Notes Statis. 85 (1994). | MR | Zbl
[7] , Models inequalities and limit theorems for stationary sequences, in Theory and applications of long range dependence, Doukhan et al. Ed., Birkhäuser (2003) 43-101. | MR | Zbl
[8] and , Rates in the empirical central limit theorem for stationary weakly dependent random fields. Stat. Inference Stoch. Process. 5 (2002) 199-228. | MR | Zbl
[9] and , A new weak dependence condition and applications to moment inequalities. Stoch. Proc. Appl. 84 (1999) 313-342. | MR | Zbl
[10] , and , Weak dependence for infinite ARCH-type bilinear models. Statistics 41 (2007) 31-45. | MR | Zbl
[11] , and , Vector valued ARCH() processes, in Dependence in Probability and Statistics, P. Bertail, P. Doukhan and P. Soulier Eds. Lecture Notes in Statistics, Springer, New York (2006). | MR | Zbl
[12] and , An invariance principle for weakly dependent stationary general models. Prob. Math. Stat. 27 (2007) 45-73. | MR | Zbl
[13] and , ARCH-type bilinear models with double long memory. Stoch. Proc. Appl. 100 (2002) 275-300. | MR | Zbl
[14] and , Goodness-of-fit tests for Markovian time series models. Technical Report No. 16/2005. Department of Mathematics and Statistics, University of Cyprus (2005).
[15] , Limit theorems of probability theory. Clarendon Press, Oxford (1995). | MR | Zbl
[16] , Nonparametric functional estimation. Academic Press, New York (1983). | MR | Zbl
[17] , About the Lindeberg method for strongly mixing sequences. ESAIM: PS 1 (1997) 35-61. | EuDML | Numdam | MR | Zbl
[18] , Théorie asymptotique pour des processus aléatoires faiblement dépendants. SMAI, Math. Appl. 31 (2000). | MR | Zbl
[19] , Nonparametric estimators for time series. J. Time Ser. Anal. 4 (1983) 185-207. | MR | Zbl
[20] , Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 (1975) 237-302. | MR | Zbl
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