Dependent Lindeberg central limit theorem and some applications
ESAIM: Probability and Statistics, Tome 12 (2008), pp. 154-172.

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.

DOI : https://doi.org/10.1051/ps:2007053
Classification : 60F05,  62G07,  62M10,  62G09
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/}
}
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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] D. Andrews, Non strong mixing autoregressive processes. J. Appl. Probab. 21 (1984) 930-934. | MR 766830 | Zbl 0552.60049

[2] P. Billingsley, Convergence of Probability Measures. Wiley, New-York (1968). | MR 233396 | Zbl 0172.21201

[3] A.V. Bulinski and A.P. Shashkin, Rates in the central limit theorem for weakly dependent random variables. J. Math. Sci. 122 (2004) 3343-3358. | MR 2078752 | Zbl 1071.60013

[4] A.V. Bulinski and A.P. Shashkin, Strong Invariance Principle for Dependent Multi-indexed Random Variables. Doklady Mathematics 72 (2005) 503-506. | MR 2161593 | Zbl 1137.60015

[5] C. Coulon-Prieur and P. Doukhan, A triangular central limit theorem under a new weak dependence condition. Stat. Prob. Letters 47 (2000) 61-68. | MR 1745670 | Zbl 0956.60006

[6] P. Doukhan, Mixing: Properties and Examples. Lect. Notes Statis. 85 (1994). | MR 1312160 | Zbl 0801.60027

[7] P. Doukhan, 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 1956044 | Zbl 1032.62081

[8] P. Doukhan and G. Lang, Rates in the empirical central limit theorem for stationary weakly dependent random fields. Stat. Inference Stoch. Process. 5 (2002) 199-228. | MR 1917292 | Zbl 1061.60016

[9] P. Doukhan and S. Louhichi, A new weak dependence condition and applications to moment inequalities. Stoch. Proc. Appl. 84 (1999) 313-342. | MR 1719345 | Zbl 0996.60020

[10] P. Doukhan, H. Madre and M. Rosenbaum, Weak dependence for infinite ARCH-type bilinear models. Statistics 41 (2007) 31-45. | MR 2303967 | Zbl 1125.62100

[11] P. Doukhan, G. Teyssiere and P. Winant, 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 2283258 | Zbl 1113.60038

[12] P. Doukhan and O. Wintenberger, An invariance principle for weakly dependent stationary general models. Prob. Math. Stat. 27 (2007) 45-73. | MR 2353271 | Zbl 1124.60031

[13] L. Giraitis and D. Surgailis, ARCH-type bilinear models with double long memory. Stoch. Proc. Appl. 100 (2002) 275-300. | MR 1919617 | Zbl 1057.62070

[14] M.H. Neumann and E. Paparoditis, Goodness-of-fit tests for Markovian time series models. Technical Report No. 16/2005. Department of Mathematics and Statistics, University of Cyprus (2005).

[15] V. Petrov, Limit theorems of probability theory. Clarendon Press, Oxford (1995). | MR 1353441 | Zbl 0826.60001

[16] B.L.S. Prakasha Rao, Nonparametric functional estimation. Academic Press, New York (1983). | MR 740865 | Zbl 0542.62025

[17] E. Rio, About the Lindeberg method for strongly mixing sequences. ESAIM: PS 1 (1997) 35-61. | EuDML 104240 | Numdam | MR 1382517 | Zbl 0869.60021

[18] E. Rio, Théorie asymptotique pour des processus aléatoires faiblement dépendants. SMAI, Math. Appl. 31 (2000). | MR 2117923 | Zbl 0944.60008

[19] P.M. Robinson, Nonparametric estimators for time series. J. Time Ser. Anal. 4 (1983) 185-207. | MR 732897 | Zbl 0544.62082

[20] M.S. Taqqu, Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 (1975) 237-302. | MR 400329 | Zbl 0303.60033

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